[size=150]Each expression represents two numbers. Evaluate the expressions and find the two numbers.[/size][br][br][math]1\pm\sqrt{49}[/math]

[math]\frac{8\pm2}{5}[/math]

[math]\pm\sqrt{(\text{-}5)^2-4\cdot4\cdot1}[/math]

[math]\frac{\text{-}18\pm\sqrt{36}}{2\cdot3}[/math]

[math]x^2-2x-1.25=0[/math]

[math]5x^2+9x-44=0[/math]

[math]x^2+1.25x=0.375[/math]

[math]4x^2-28x+29=0[/math]

[size=150]Here is a formula called the [b]quadratic formula[/b].[br][center][math]x=\frac{\text{-}b\pm\sqrt{b^2-4ac}}{2a}[/math][/center]The formula can be used to find the solutions to any quadratic equation in the form of [math]ax^2+bx+c=0[/math], where [math]a[/math], [math]b[/math], and [math]c[/math] are numbers and [math]a[/math] is not 0.[br][br]This example shows how it is used to solve [math]x^2-8x+15=0[/math], in which [math]a=1[/math], [math]b=\text{-}8[/math], and [math]c=15[/math].[br][br][size=100][math]\displaystyle \begin {align} x &=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a} &\qquad &\text{original equation}\\ x &=\dfrac{\text- (\text-8) \pm \sqrt{(\text-8)^2-4(1)(15)}}{2(1)} &\qquad &\text{substitute the values of }a, b, \text{and }c &\\ x &=\dfrac{8 \pm \sqrt{64-60}}{2} &\qquad &\text {evaluate each part of the expression} \\ x &=\dfrac{8 \pm \sqrt{4}}{2} \\x &=\dfrac{8 \pm 2}{2}\\ x &=\frac{10}{2} \qquad \text {or} \qquad x =\frac{6}{2} \\x &=\text{ }5 \qquad \text{ } \text {or} \qquad x =\text{ }3 \end{align}[/math][/size][br][br]Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct.[br][/size][br][math]x^2+4x-5=0[/math]. The solutions are [math]x=\text{-}5[/math] and [math]x=1[/math].[br][br]

[math]x^2+7x+12=0[/math]. The solutions are [math]x=\text{-}3[/math] and [math]x=\text{-}4[/math].

[math]x^2+10x+18=0[/math]. The solutions are [math]x=\text{-}5\pm\frac{\sqrt{28}}{2}[/math].

[math]x^2-8x+11=0.[/math] The solutions are [math]x=4\pm\frac{\sqrt{20}}{2}[/math].

[math]9x^2-6x+1=0[/math].  The solution is [math]x=\frac{1}{3}[/math].

[math]6x^2+9x-15=0[/math]. The solutions are [math]x=\text{-}\frac{5}{2}[/math] and [math]x=1[/math].

[size=150]Use the quadratic formula to solve [math]ax^2+c=0[/math]. Let’s call the resulting equation P.[/size]

[size=100]Solve the equation [math]3x^2-27=0[/math] in two ways, showing your reasoning for each:[/size][br][list][*]Without using any formulas.[/*][/list]

[list][*]Using equation P.[br][/*][/list]

Check that you got the same solutions using each method.[br][br]

[size=150]Use the quadratic formula to solve [math]ax^2+bx=0[/math]. Let’s call the resulting equation Q.[/size]

[size=100]Solve the equation [math]2x^2-5x=0[/math] in two ways, showing your reasoning for each:[/size][br][list][*]Without using any formulas.[/*][/list]

[list][*]Using equation Q.[br][/*][/list]

Check that you got the same solutions using each method.

[size=150]For each equation, identify the values of [math]a[/math], [math]b[/math], and [math]c[/math] that you would substitute into the quadratic formula to solve the equation.[/size] [br][br][math]3x^2+8x+4=0[/math]

[math]2x^2-5x+2=0[/math]

[math]\text{-}9x^2+13x-1=0[/math]

[math]x^2+x-11=0[/math]

[math]\text{-}x^2+16x+64=0[/math]

[math]x^2+9x+20=0[/math]. The solutions are [math]x=\text{-}4[/math] and [math]x=\text{-}5[/math].

[math]x^2-10x+21=0[/math]. The solutions are [math]x=3[/math] and [math]x=7[/math].

[math]3x^2-5x+1=0[/math]. The solutions are [math]x=\frac{5}{6}\pm\frac{\sqrt{13}}{6}[/math].

[size=150]Select [b]all[/b] the equations that are equivalent to [math]81x^2+180x-200=100[/math].[/size]

Which object reaches the ground first? Explain how you know.

What is the maximum height of each object?

[size=150]Identify the values of [math]a[/math], [math]b[/math], and [math]c[/math] that you would substitute into the quadratic formula to solve the equation.[br][/size][br][math]x^2+9x+18=0[/math][br]

[math]4x^2-3x+11=0[/math]

[math]81-x+5x^2=0[/math]

[math]\frac{4}{5}x^2+3x=\frac{1}{3}[/math]

[math]121=x^2[/math]

[math]7x+14x^2=42[/math]