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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[/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].
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Information: IM Alg1.7.15 Lesson: Quadratic Equations with Irrational Solutions