2.EE: Exercises for Quadratic Equations
Solving Quadratic Equations Using the Square Root Property
In the following exercises, solve using the Square Root Property.
- \(y^{2}=144\)
- \(n^{2}-80=0\)
- \(4 a^{2}=100\)
- \(2 b^{2}=72\)
- \(r^{2}+32=0\)
- \(t^{2}+18=0\)
- \(\frac{2}{3} w^{2}-20=30\)
- \(5 c^{2}+3=19\)
- Answer
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1. \(y=\pm 12\)
3. \(a=\pm 5\)
5. \(r=\pm 4 \sqrt{2} i\)
7. \(w=\pm 5 \sqrt{3}\)
In the following exercises, solve using the Square Root Property.
- \((p-5)^{2}+3=19\)
- \((u+1)^{2}=45\)
- \(\left(x-\frac{1}{4}\right)^{2}=\frac{3}{16}\)
- \(\left(y-\frac{2}{3}\right)^{2}=\frac{2}{9}\)
- \((n-4)^{2}-50=150\)
- \((4 c-1)^{2}=-18\)
- \(n^{2}+10 n+25=12\)
- \(64 a^{2}+48 a+9=81\)
- Answer
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1. \(p=-1,9\)
3. \(x=\frac{1}{4} \pm \frac{\sqrt{3}}{4}\)
5. \(n=4 \pm 10 \sqrt{2}\)
7. \(n=-5 \pm 2 \sqrt{3}\)
Solving Quadratic Equations by Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
- \(x^{2}+22 x\)
- \(m^{2}-8 m\)
- \(a^{2}-3 a\)
- \(b^{2}+13 b\)
- Answer
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1. \((x+11)^{2}\)
3. \(\left(a-\frac{3}{2}\right)^{2}\)
In the following exercises, solve by completing the square.
- \(d^{2}+14 d=-13\)
- \(y^{2}-6 y=36\)
- \(m^{2}+6 m=-109\)
- \(t^{2}-12 t=-40\)
- \(v^{2}-14 v=-31\)
- \(w^{2}-20 w=100\)
- \(m^{2}+10 m-4=-13\)
- \(n^{2}-6 n+11=34\)
- \(a^{2}=3 a+8\)
- \(b^{2}=11 b-5\)
- \((u+8)(u+4)=14\)
- \((z-10)(z+2)=28\)
- Answer
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1. \(d=-13,-1\)
3. \(m=-3 \pm 10 i\)
5. \(v=7 \pm 3 \sqrt{2}\)
7. \(m=-9,-1\)
9. \(a=\frac{3}{2} \pm \frac{\sqrt{41}}{2}\)
11. \(u=-6 \pm 2 \sqrt{2}\)
Solving Quadratic Equations of the Form \(ax^{2}+bx+c=0\) by Completing the Square
In the following exercises, solve by completing the square.
- \(3 p^{2}-18 p+15=15\)
- \(5 q^{2}+70 q+20=0\)
- \(4 y^{2}-6 y=4\)
- \(2 x^{2}+2 x=4\)
- \(3 c^{2}+2 c=9\)
- \(4 d^{2}-2 d=8\)
- \(2 x^{2}+6 x=-5\)
- \(2 x^{2}+4 x=-5\)
- Answer
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1. \(p=0,6\)
3. \(y=-\frac{1}{2}, 2\)
5. \(c=-\frac{1}{3} \pm \frac{2 \sqrt{7}}{3}\)
7. \(x=\frac{3}{2} \pm \frac{1}{2} i\)
In the following exercises, solve by using the Quadratic Formula.
- \(4 x^{2}-5 x+1=0\)
- \(7 y^{2}+4 y-3=0\)
- \(r^{2}-r-42=0\)
- \(t^{2}+13 t+22=0\)
- \(4 v^{2}+v-5=0\)
- \(2 w^{2}+9 w+2=0\)
- \(3 m^{2}+8 m+2=0\)
- \(5 n^{2}+2 n-1=0\)
- \(6 a^{2}-5 a+2=0\)
- \(4 b^{2}-b+8=0\)
- \(u(u-10)+3=0\)
- \(5 z(z-2)=3\)
- \(\frac{1}{8} p^{2}-\frac{1}{5} p=-\frac{1}{20}\)
- \(\frac{2}{5} q^{2}+\frac{3}{10} q=\frac{1}{10}\)
- \(4 c^{2}+4 c+1=0\)
- \(9 d^{2}-12 d=-4\)
- Answer
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1. \(x=\frac{1}{4}, 1\)
3. \(r=-6,7\)
5. \(v=\frac{-1 \pm \sqrt{21}}{8}\)
7. \(m=\frac{-4 \pm \sqrt{10}}{3}\)
9. \(a=\frac{5}{12} \pm \frac{\sqrt{23}}{12} i\)
11. \(u=5 \pm \sqrt{21}\)
13. \(p=\frac{4 \pm \sqrt{5}}{5}\)
15. \(c=-\frac{1}{2}\)
In the following exercises, determine the number of solutions for each quadratic equation.
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- \(9 x^{2}-6 x+1=0\)
- \(3 y^{2}-8 y+1=0\)
- \(7 m^{2}+12 m+4=0\)
- \(5 n^{2}-n+1=0\)
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- \(5 x^{2}-7 x-8=0\)
- \(7 x^{2}-10 x+5=0\)
- \(25 x^{2}-90 x+81=0\)
- \(15 x^{2}-8 x+4=0\)
- Answer
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1.
- \(1\)
- \(2\)
- \(2\)
- \(2\)
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.
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- \(16 r^{2}-8 r+1=0\)
- \(5 t^{2}-8 t+3=9\)
- \(3(c+2)^{2}=15\)
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- \(4 d^{2}+10 d-5=21\)
- \(25 x^{2}-60 x+36=0\)
- \(6(5 v-7)^{2}=150\)
- Answer
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1.
- Factor
- Quadratic Formula
- Square Root
Solving Equations in Quadratic Form
In the following exercises, solve.
- \(x^{4}-14 x^{2}+24=0\)
- \(x^{4}+4 x^{2}-32=0\)
- \(4 x^{4}-5 x^{2}+1=0\)
- \((2 y+3)^{2}+3(2 y+3)-28=0\)
- \(x+3 \sqrt{x}-28=0\)
- \(6 x+5 \sqrt{x}-6=0\)
- \(x^{\frac{2}{3}}-10 x^{\frac{1}{3}}+24=0\)
- \(x+7 x^{\frac{1}{2}}+6=0\)
- \(8 x^{-2}-2 x^{-1}-3=0\)
- Answer
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1. \(x=\pm \sqrt{2}, x=\pm 2 \sqrt{3}\)
3. \(x=\pm 1, x=\pm \frac{1}{2}\)
5. \(x=16\)
7. \(x=64, x=216\)
9. \(x=-2, x=\frac{4}{3}\)