IM Alg1.7.15 Lesson: Quadratic Equations with Irrational Solutions

Here are some squares whose vertices are on a grid.
Find the area and the side length each square.
[size=150]Solve each equation. Use the [math]\pm[/math] notation when appropriate.[/size][br][br][math]x^2-13=-12[/math]
[math]\left(x-6\right)^2=0[/math]
[math]x^2+9=0[/math]
[math]x^2=18[/math]
[math]x^2+1=18[/math]
[math]\left(x+1\right)^2=18[/math]
Here is an example of an equation being solved by graphing and by completing the square.
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JhcDEICJ0VLo8PNBi84n9v0GIB7naFCxyvSi9B5hqPTeVGQbgPgNHyK5AEK+MpKQfHuFaniIAQjp8fcdPc2+++7b+zY/rOl5KgMIjRWe+D2+z4tSAKFDACCYNnzhC1+ogMEHEGSHBk34j3/846oD8UcDMkKiPoUOd38EQXljAOJ3Vv43Ic/+90oefhyeDUA8jtQx04tSPVJRNDRBfGnITSMQv+FDhLz8CvLz8MsBbZnCUImxfP20dc+pAILgvOc97+k8jZmqACINp2lUVgIgyBD1TT5cyAmd2P7779/rKERe8Kf+GTUiF+JIDwgxfcIf4MCP+CKjxC0BEOjCA3+qA00DEKmFnY3aH6b985//rFAf5Oc666yzqgrxh48kh7lNI5Cw4ZOGCuaKOfylHNAGQBAE8vV7Fck3RiP06wIgxx9/fOdpzFQFEEaZohCF3+LYlSpyceihh1ab0ySMO3HheY5jdEFaGWlypwwCBkwtAAfkChcDC/xJ48tYCYBAz5ehKmMDkP45o99wYdB//vOfamcmuzO5qADi0JBFeLjTI+BHhf385z8X3vbugAE9hJ+Gd/HzQYFEfjkEQJryha4/pO1l7D1Q+SJwvg6EMvuOspIX/jJF88PrnqcqgFCvNNiw8axfv74nG/CL3cK+KwEQ5IH8PvvZz1agwXPY84cy4uuypBxSl/IeAxDkBjkMHfXv60BEz+KDKGlsBOJxDobFmOlFqXoWKs+/aGgMLT/5yU9W5yz8+DxTSX58qXyp4DbhYARCxYU0JF/8w4oNy8DwVno0ARD8QpCQMsELGk+qm4oAAuBK46V+4XPMlUxhYvQkX+pLGq6MPiQ+9UNd4SROKEchCMQAJJZW/PyOrQ4QDUCkRrxhZ1hZXpToI8ylMkUH8uSTT7rXvOY17ktf+lI0Pp4CBrEIhIlwQFt0IGFcydf3R+BlpOH7iwBw6IsDUN/5zncm9KrEl/QiRDFaPl15nooAAsBy4QDaEGzl27sAyPz5893LXvay3oiFuqZOkDkungFufyMZQMBGMpFL6pKOzgc0wIZRCPVGPDaa+ToQ4u6+++7RjWR8F2kBDC6eRf7kGwEsQDR0BiABR2B623QgSFIxnQoUAPnXv/7lnvvc57pPfOITYdTeO3nU5YO/jBaaAITKlngQFpAQQetltvMBuggBIPGKV7xiAtAAFr7QdZnGTDUAgYfwSeqIe92ILAYg1E3YCKkGGiJ0mQLhoAuf8eMiDX7hKgyNnLohDvF98NhZvZUsSP0CfJRBHO8f+MAHogDCt0JfykDcUIYoVwxAxxZAON770EMPCX97dyomNp/sRWh44Fjz3//+9yrGE0880RAzPYjj15waTnGUXXrMpvj0clu2TDy/gJD4gIQgI1ShMMVoc7qZE76ajqP8nMbVdJgx8BtWHe3Yt9M48Q8dJ5H//Oc/h9617yjn25wPIG1xU8IpO/LBqdSuTkaj3EPHfyzTeWg6OmKm2ZqOzvWGG25oJZms9sZ+RQxAaCwwO3Xo7pcI+xUCIL5/yTP2K7oACIxqc9iFCAGEdHy3DxY8AyCxRhPmgQ0HbQDBJoQ2gFCvKQCCYjrscUkb8gg+AJ5dACTkXexdE0CoRzoWzDjkAAgdS2zEQ7kxuaANIIDSyAIITAFpU4QsrPhdDSBheereYwDC98Z6GASPERl2SZrcVAGQBx980P3sZz+rgDPWiQC0PsjCk2EHEKm3XABp6pSGEUDq6m1SRiDC7Jz7KANI3ffOmjWrNzdGMVw3GpoKAPKLX/yi962MvL797W/XsaXPf6oDSN/HBi/DCCB0howU2XohDhBUBZCf/OQn1Z4NzNFpXaeddpqDrhY96Hzuc5+rFG+aNOfMmePOO++81nKefPLJfQ2KRnXMMcdE033+8593P/jBD6JhuWVnW/f3v/99VZqsjH33u9+N0mSVgm/0r3nz5kXj+t905plnunPPPbc1nqT5xje+4U4//fTG6+Uvf3ll/UzSaNy/8pWvVKCoQUtosKGO6Y28a9y//vWvO3hUQuuEE06o6pHVKoBEHUC++MUvumc/+9mq17Oe9SzHpUl3V9J85jOf2deYaFjPeMYzot+3K8vZhd9N5fSBQ55T6rOJZqxs8FDoN93reB2jmeLXtZyjTtPnLfos1RFInRJVhjw596k2hWHFwq8EnhllxNxUmMLMnj2773tf/epXxz51gt8gpjAvfOELs1cDJxRwp0euDqSOHv7DOIWhXCh+kVeAA92e+gjEAGTiMm5MUAAMhoH0XgcffHAsSuU3FQAEQ77Pec5zquvtb3978uqCAYjuMm7pKgxTKgEOEVgDEOGEwj22CtNGNrYvwk8zFQCk7Rv97/WfuwAICj5ZEiU/OR8Vmii0EUjePhBWyGKriQYgvsQWPucACBXDTkh2UsbcVAAQlqvDvR+xbw39UgEEwGCjnuzLQKgRdgCF4bYAC/QNQPIAJKwbeVcHEHYP7tgx0TK5ZJhzZz64ffv2nKS1abAev23bttrwnAB2+m3durVzUhpX3WFDaQydiTYk4H9M2Jeh6dj0FDMszMiARsx3dHX8j0m4MS+kIeCBP3trwr0KzNn9rfKDAJB169a5zZs3h0UremcDnfZmPw79sUVe01G/0G1zyTtR7W8duoNnUyMb9bMwgCNnTHJc7CyMTwe+sS8BB3AAFqGTMzLiPwgAiZ3Glfxy72N7FsYApDuAIGQIf+y8zSgDCFMKRh+MEnJcG4DAL6HNCIf8Qoc/ZRBnAHKvsELlzsjL/plOhZVPG1XOIUdDQNBDRdUoAwi6B/8Ucle+NAEIfIrxK8wDvsoohTADkCEHkLlz51Y2F+hRqbhQCx5WcMq7HOdPiZsah56JP8NGwUc5c4fZfn5iUMj36/JMOfxTu6T1AYSw2DC9Sx7ERbcQO/DYlY4fn52tBx10UMVHUQqj1wm/x0/T9twEIOHIoo4WUyi/DMMOIHwXK0ive93rKoDUaD/CG3Qq99475ACC0VqOi+MYUtIomKuWuEEACHNjbHeUls3/rlIAocdG4ecPxQVAECyG7MMKIJ/61KcqYJJ6f/GLX5w0QvD5Fz43AYiM2MI0/rvIn89PzdO4kpemDgS5ZHSFDoQ/7Kb9hIphybfrvQRARE8nvORO54tMqk5hdtttt76j9wg8wl/iBgEgb33rW91Pf/rTkmJNSFsKIFQKvbfM68kAAMEuqAjSsAIIm8X8FSjOv7zzne+cwKMuHk0AIiOQpsbFyMPnJXkPO4AIf0SJSqfiL0NLeM69BEDID9mT7QaM7AAPdR3Il7/8ZTdz5swKNRl+aQj8IACE+bMY8KWcoe4hp4JKAYQ8wyVdAIS/g6Ch0Gg0+DmIKYwPINK4ly9fnsPGXpomABGwZbgfcwBHCB7EGzUAYWqtNUouBRBkkGkpIEK5qAN1ACETKokek+G4xscPCkA++tGPVg0TQaPnLx0paQCIKAdF+DlB+ZGPfKRqI6MCIEwNOfVa6poABNrwiI6AoTTW4OAdd0YedSOTUQIQ6p5eXsuVAgjloF3TVqTDLQIQTA0effTR1SWGjtGBcKwbh8CTYWnDLAUQekIp56mnnlqVDcHzNz4hjPT+Ja4UQNAdUU6MAzMVpJKwFSJ/bTAsAMJGNOEn9kBxMgLB1ge8La1zaLYBCHHIB8BgZEZjE+CtChX5GRUA4U/aDzzwwD59WORzOnmVAgiDAQYF/qbHIgDBBuVdd91VXWj1qUwAhKGNOOZvpQ2zFECYqvgXZUPIfavsGo2zFECkjAAJ5WOExD12lTTQ0imMlFPu8BMA+eEPf1j9iThTWA2XAiBd8xkFAAEEWYVBiarpSgCENg1wUDbRx1G2IgAJP44e86UvfWmfEhXwEMVLGD/1vRRAYvmgV+AvGMRplLMUQKQs3OlNmd/LKgx+GiAHnVIA8cspzxiqYbQE2LWNAiRN230cAQTeoV+46aabWrfxt/EvDM8FEMCDaaIshzMoEN2TKoBQYCwWzZgxo8qATBla+iOS8KNS3gcBIPTyHKOHEVrl1AQQWTZjpUjOAQ0zgAAcDG+f//znVzyFryJwKXUcizOOAAIfkcfDDz/cvfnNb654WcpH4W0ugNC5+sp72jN6EGRUHUBkKzvCXgoc8uGDABDKxxSGUZMohCS/3LsmgFAGKm769Ok9AMktV5huECMQ7MAg/HXKy7AMKe/jCCDCF1nGlXeNey6ANOWtDiAIEEo2Kl/rwiIZDV6LHnT4WwdMsWnSxNYkYKdFk+PsNEqWxrVoQgezA2vXrlWlyWgOQ9Ga5eQ/hjjdXUITHvrXG97whuqPoHw/nkvyYDEBK3MlNMK0tCMs9of+Je+sUDHyLqERpoV3qhvJ+HCQTnp2jTt/AoWyVoOW0KCcGzZsUKUJM9mVKHlo3N/1rne5fffdV5UmgomCTqN80GCVC6A7//zz1WhC99prr3V/+tOfimhyLN6/mB587GMf6/MjvIQXjLrRWZTQCNPSEaH/Cv1L3umEGSWX0AjTMlhQBRCZwjQNe7qGDXIK07UsTfG1pzDkhUGhPfbYQ203IjS1pzA0yje96U19O1Gb+JQaRm+HwGq6UViF4XvHdgpjAJJ3nL+ukdAL8fcG4RmZuvgp/poAwqqB6D78rewp5WiLYwCSZl+3jY8SPhI6EAMQfQCRszAoVTWcFoCgJAfYWCWQjWQa5RMaBiAGICILRfdxnsLIPhB0NvT0zGNLnRaAABws5wEkBiD3l1ZLX3qbwvSxo+zFAOQpe7CsxfvbiHO5qgEgAJlMXSiHAYgBSJM8Pm0TrimWc86mMPpTGNlIhkKRHr/0eHcpgDDiAMj8g14GIAYgTdBgANLEnZ1hg1iFkSmMZM+xgNKpTCmAyNTFXyExADEAERmN3Q1AYlwJ/CYDQMiSnp8RQO5O3xIAEV0Md98ZgBiA+PIQPhuAhByJvE8WgAAcTGWOO+64SCnavXIBhBEHqy6x1SADEAOQJskzAGnizs6wyQIQshMlZs5J5xwAAbTYMCaWqEJ2GIAYgIQy4b8bgPjcqHmeTAChCLKJq+vx+RwAOemkk3onMGOfbwBiABKTC/EzABFONNwnG0AoCtMJphVdTEd2BRDAA8VtUx4GIAYgDU3DGYA0cWdn2K4AkBwQ6QIgAh5toxwDEAOQpiZiANLEnV0MID6IhKsjsWKnAkgqeJCHAYgBSEzWxM8ARDjRcN9VIxApEtMZphpt/2bWBiAoTLEoxkpPCiCR/2QDSO4Stp3GHfJ/prOdqIPbiSpA0XQXxSoA4G/08tM0AQiAgU4Fw7lNOg+fHs+TBSDswqVsbOvn3nUVygDEACSU3ax3lkF9q+xZRIJEu3oEIsWh4bPcymiEaUgIJDEAwVoVoEMadpp27eEnC0B8XQxl7Ap0BiBDDiBYkaJh/u9//1O7ML+3Y8cONXqU7brrrnMPP/ywKk1MJG7btk2VJhafsLORw8/58+dXf3QOKAAoWE6//PLLq7/0xKzhhRde6ObMmVM1QuIceeSR1VmmnLwApQcffDCrnHX5YY2M/+6pC8e/rcxPPPGE86/Zs2e70047rc+P8KY82sIwO8CfVrfF6xLOX2Zgb7RLmra4dCzYBGmL1yWcU8OqFskYAmNFix5N68JU3qWXXqpGj3LNnTu3sg2qVUbozJs3rzLvp00TW6slNL/2ta85TCPus88+1QgDsJALW6b8OdQFF1xQlAdghZ3ZknKGaQG4pUuX1tIkP76pKV/qhLqWi7+egBfyzn3hwoW1eYRlir0vWLDAYVQ6Fpbrx0hrEDQXL16sWs5FixbpAojpQHatDkSmMU332BSmKX5K2GRNYfyycCbIdCA+R5qfzSJZM386hU5lHUgbI0YFQPiL1AMOOKD6i8dTTjml77NYcYqdyemLFHkxHciQ60BsBGIjkEi7zfJCn4b1ePRfXOJywYP0BiAGICJHRXcbgTxUxL8w8WRNYUrAgzIbgBiAhLKb9W4AMnoAQp2J8te/+3+72CYMBiAGIG0ykhRuADJ6AJJUsS2RDEAMQFpEJC3YAMQAJNU9BRUAAASWSURBVE1S2mPxb4TsV9F0ZpVdkZtmlf0pq+xaLB2VVRj7Xxj7XxgVmTcAMQBRESRTolY7ZrV4CR12y6ruRLVl3KeXGzUqKrTKrkHTRiBnaLCxR8OmMHf3eFH3YMf56zjj+Q/LYTqvSNFHAxADkKhgZHjaCCSDaXVJDEC21rEmy990IKYDyRKcMJHpQEwHEspE7rst49oybq7s9KWzZVxbxu0TiIIX04GYDqRAfJ5OalMYm8I8LQ1lT7YPpIx/faltCmNTmD6BKHixKYxNYQrE5+mkNoWxKczT0lD2ZFMYm8KUSdDO1DaFsSmMiiA558Z2CoNJw3Xr1lW709ihpnFhv5Pj4hq0hAZmF2+55RZVmpdddpljj4XkoXHHhimbyTRoCY0rr7zS3Xjjjao0qXfst0oeGvcVK1ZU/wFcQovpr3/NnDnTfehDH+rzI7wkj6uuuqqysVtCI0y7atWqyj5t6F/yvnr1ardmzZqibw3zx2YLfm0ueSMZgoQRmHvuuUftwh4qewI0adLYASVNmjR2QEmT5hVXXFGBkiZN6ghQ0qRJY2cEpklz5cqVFSiV0ATU/GvWrFnu+OOP7/MjvCSPa665xmFQu4RGmJbGjuHv0L/kncbOTvESGmFaASQ1ALGt7LaVvU2YUsNtI5ltJEuVlcZ4tgpjqzCNAtIh0FZhbBWmg7jUR7VVGFuFqZeObiG2CqOoA7EpjE1hujW/+tg2hbEpTL10dAixKYxNYTqIS2NUm8LYFKZRQFIDbQpjU5hUWWmLZ1MYm8K0yUhSuG0ks41kSYKSEGlsN5KZDsR0IAntIymK6UBMB5IkKG2RTAdiOpA2GUkNNx2I6UBSZaUxnulATAfSKCAdAk0HYjqQDuJSH9V0IKYDqZeObiGmA+nGr8bYNoWxKUyjgHQItCmMTWE6iEt9VJvC2BSmXjq6hdgUxqYw3SSmJrZNYWwKUyManb1tCtOZZfUJbApjU5h66egWYlMYm8J0k5ia2DaFGf0pzIknnuiOOuqomhqOexuAGIDEJaOjrwHIaAMIRpH2228/A5AOcr9p0yb7b9wO/GqMagAyugDy+OOPV+ABiNgIpFHM+wJHAkAwD4cNU+xual0LFixwy5cvV6NHuaB5ySWXjATNZcuWqZdz6dKl6jQvvvhidZpLliyZQPPggw9255xzTnXx3CRnc+fOdeeff37vOvbYYx2X74csNNFoCyP94sWLi2iEeUBz0aJFqjQXLlzoLrroIlWa0FO1ifrkk086u4wHg5KB8847z82ZM6eSMWyGHnnkkY3y9t///telXIMq7zjQ7Rvm1LwkG1WuSW/exoHOHMBy/tFHH11dp5xyirvzzjurKQtTGBzT0K5TmM6FsAQqHDAAUWGjEenCgfXr1zv/Ovvss920adOiVxe6FnfyOWAAMvk8txxbOGAjkBYGDVGwAcgQVYYV5SkOGICMjiQYgIxOXVlJjQNDxwEDkKGrEiuQcWB0OGAAMjp1ZSU1DgwdBwxAhq5KrEDGgdHhgAHI6NSVldQ4MHQcMAAZuiqxAhkHRocDBiCjU1dWUuPA0HHg/yjOgIuO2v5DAAAAAElFTkSuQmCC[/img][/td][td][math]\displaystyle \begin {align} x^2 + 6x +7 &=0\\ x^2 + 6x + 9 &= 2\\(x+3)^2 &= 2\\x+3 &=\pm \sqrt2\\ x &=\text-3\pm \sqrt2 \end{align}[/math][br][br]Verify: [math]\sqrt{2}[/math] is approximately 1.414. [br]So [math]\text{-}3+\sqrt{2}\approx\text{-}1.586[/math] and [math]\text{-}3-\sqrt{2}\approx\text{-}4.414[/math].[/td][/tr][/table][br]For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. If you get stuck, study the example.[/size][br][br][math]x^2+4x+1=0[/math]
[math]x^2-10x+18=0[/math]
[math]x^2+5x+\frac{1}{4}=0[/math]
[math]x^2+\frac{8}{3}x+\frac{14}{9}=0[/math]
Write a quadratic equation of the form [math]ax^2+bx+c=0[/math] whose solutions are [math]x=5-\sqrt{2}[/math] and [math]x=5+\sqrt{2}[/math].

IM Alg1.7.15 Practice: Quadratic Equations with Irrational Solutions

Solve each equation and write the solutions using ± notation.
[math]x^2=144[/math]
[math]x^2=5[/math]
[math]4x^2=28[/math]
[math]x^2=\frac{25}{4}[/math]
[math]2x^2=22[/math]
[math]7x^2=16[/math]
Match each expression to an equivalent expression.
Is [math]\sqrt{4}[/math] a positive or negative number? Explain your reasoning.[br]
Is [math]\sqrt{5}[/math] a positive or negative number? Explain your reasoning.[br]
Explain the difference between [math]\sqrt{9}[/math] and the solutions to [math]x^2=9[/math].[br]
For each equation, find the exact solutions by completing the square and the approximate solutions by graphing.
[size=150]Then, verify that the solutions found using the two methods are close.[/size][br][br][math]x^2+10x+8=0[/math]
[math]x^2-4x-11=0[/math]
Jada is working on solving a quadratic equation, as shown here.
[table][tr][td][br][math]\begin{align} p^2-5p&=0\\p(p-5)&=0\\p-5&=0\\p&=5\end{align}[/math][br][/td][td][size=150]She thinks that her solution is correct because substituting [br]5 for [math]p[/math] in the original expression [math]p^2-5p[/math] gives [math]5^2-5\left(5\right)[/math], [br]which is [math]25-25[/math] or 0.[/size][/td][/tr][/table][br]Explain the mistake that Jada made and show the correct solutions.
[size=150]Which expression in factored form is equivalent to [math]30x^2+31x+5[/math]?[/size]
[size=150]Two rocks are launched straight up in the air. The height of Rock A is given by the function [math]f[/math], where [math]f\left(t\right)=4+30t-16t^2[/math]. The height of Rock B is given by [math]g[/math], where [math]g\left(t\right)=5+20t-16t^2[/math]. In both functions, [math]t[/math] is time measured in seconds and height is measured in feet.[/size][br][br]Which rock is launched from a higher point?
Which rock is launched with a greater velocity?
Describe how the graph of [math]f(x)=|x|[/math] has to be shifted to match the given graph.[br][img]data:image/png;base64,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[/img]
Find an equation for the function represented by the graph.

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