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Determine the number of solutions to each quadratic equation:
ⓐ $8{m}^{2}-3m+6=0$ ⓑ $5{z}^{2}+6z-2=0$ ⓒ $9{w}^{2}+24w+16=0$ ⓓ $9{u}^{2}-2u+4=0$
ⓐ no real solutions ⓑ 2 ⓒ 1 ⓓ no real solutions
Determine the number of solutions to each quadratic equation:
ⓐ ${b}^{2}+7b-13=0$ ⓑ $5{a}^{2}-6a+10=0$ ⓒ $4{r}^{2}-20r+25=0$ ⓓ $7{t}^{2}-11t+3=0$
ⓐ 2 ⓑ no real solutions ⓒ 1 ⓓ 2
We have used four methods to solve quadratic equations:
You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.
What about the method of completing the square? Most people find that method cumbersome and prefer not to use it. We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.
Identify the most appropriate method to use to solve each quadratic equation:
ⓐ $5{z}^{2}=17$ ⓑ $4{x}^{2}-12x+9=0$ ⓒ $8{u}^{2}+6u=11$
ⓐ $5{z}^{2}=17$
Since the equation is in the $a{x}^{2}=k$ , the most appropriate method is to use the Square Root Property.
ⓑ $4{x}^{2}-12x+9=0$
We recognize that the left side of the equation is a perfect square trinomial, and so Factoring will be the most appropriate method.
ⓒ $8{u}^{2}+6u=11$
Put the equation in standard form. $8{u}^{2}+6u-11=0$
While our first thought may be to try Factoring, thinking about all the possibilities for trial and error leads us to choose the Quadratic Formula as the most appropriate method
Identify the most appropriate method to use to solve each quadratic equation:
ⓐ ${x}^{2}+6x+8=0$ ⓑ ${\left(n-3\right)}^{2}=16$ ⓒ $5{p}^{2}-6p=9$
ⓐ factor ⓑ Square Root Property ⓒ Quadratic Formula
Identify the most appropriate method to use to solve each quadratic equation:
ⓐ $8{a}^{2}+3a-9=0$ ⓑ $4{b}^{2}+4b+1=0$ ⓒ $5{c}^{2}=125$
ⓐ Quadratic Formula ⓑ factoring ⓒ Square Root Property
Access these online resources for additional instruction and practice with using the Quadratic Formula:
Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
$4{n}^{2}-9n+5=0$
$3{q}^{2}+8q-3=0$
${q}^{2}+3q-18=0$
${t}^{2}+13t+40=0$
$6{z}^{2}-9z+1=0$
$5{b}^{2}+2b-4=0$
$6{y}^{2}-5y+2=0$
$v\left(v+5\right)-10=0$
$v=\frac{\mathrm{-5}\pm \sqrt{65}}{2}$
$3w\left(w-2\right)-8=0$
$\frac{1}{3}{m}^{2}+\frac{1}{12}m=\frac{1}{4}$
$m=\mathrm{-1},m=\frac{3}{4}$
$\frac{1}{3}{n}^{2}+n=-\frac{1}{2}$
$25{d}^{2}-60d+36=0$
$8{n}^{2}-3n+3=0$
$25{q}^{2}+30q+9=0$
$3t\left(t-2\right)=2$
$4{d}^{2}-7d+2=0$
$\frac{3}{4}{b}^{2}+\frac{1}{2}b=\frac{3}{8}$
$b=\frac{\mathrm{-2}\pm \sqrt{11}}{6}$
$\frac{1}{9}{c}^{2}+\frac{2}{3}c=3$
$16{y}^{2}+8y+1=0$
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions to each quadratic equation.
ⓐ
$4{x}^{2}-5x+16=0$
ⓑ
$36{y}^{2}+36y+9=0$
ⓒ
$6{m}^{2}+3m-5=0$
ⓓ
$18{n}^{2}-7n+3=0$
ⓐ no real solutions
ⓑ 1
ⓒ 2
ⓓ no real solutions
ⓐ
$9{v}^{2}-15v+25=0$
ⓑ
$100{w}^{2}+60w+9=0$
ⓒ
$5{c}^{2}+7c-10=0$
ⓓ
$15{d}^{2}-4d+8=0$
ⓐ
${r}^{2}+12r+36=0$
ⓑ
$8{t}^{2}-11t+5=0$
ⓒ
$4{u}^{2}-12u+9=0$
ⓓ
$3{v}^{2}-5v-1=0$
ⓐ 1
ⓑ no real solutions
ⓒ 1
ⓓ 2
ⓐ
$25{p}^{2}+10p+1=0$
ⓑ
$7{q}^{2}-3q-6=0$
ⓒ
$7{y}^{2}+2y+8=0$
ⓓ
$25{z}^{2}-60z+36=0$
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.
ⓐ
${x}^{2}-5x-24=0$
ⓑ
${\left(y+5\right)}^{2}=12$
ⓒ
$14{m}^{2}+3m=11$
ⓐ factor
ⓑ square root
ⓒ Quadratic Formula
ⓐ
${\left(8v+3\right)}^{2}=81$
ⓑ
${w}^{2}-9w-22=0$
ⓒ
$4{n}^{2}-10=6$
ⓐ
$6{a}^{2}+14=20$
ⓑ
${\left(x-\frac{1}{4}\right)}^{2}=\frac{5}{16}$
ⓒ
${y}^{2}-2y=8$
ⓐ factor
ⓑ square root
ⓒ factor
ⓐ
$8{b}^{2}+15b=4$
ⓑ
$\frac{5}{9}{v}^{2}-\frac{2}{3}v=1$
ⓒ
${\left(w+\frac{4}{3}\right)}^{2}=\frac{2}{9}$
A flare is fired straight up from a ship at sea. Solve the equation $16\left({t}^{2}-13t+40\right)=0$ for $t$ , the number of seconds it will take for the flare to be at an altitude of 640 feet.
5 seconds, 8 seconds
An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation $\frac{1}{2}{h}^{2}+3h=140$ for $h$ , the height of the window.
Solve the equation
${x}^{2}+10x=200$
ⓐ by completing the square
ⓑ using the Quadratic Formula
ⓒ Which method do you prefer? Why?
ⓐ
$\mathrm{-20},10$
ⓑ
$\mathrm{-20},10$
ⓒ answers will vary
Solve the equation
$12{y}^{2}+23y=24$
ⓐ by completing the square
ⓑ using the Quadratic Formula
ⓒ Which method do you prefer? Why?
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
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