10.4: Solving Quadratic Equations Using the Method of Extraction of Roots

The Method Of Extraction Of Roots

Extraction of Roots

Quadratic equations of the form $x^2 - K = 0$ can be solved by the method of extraction of roots by rewriting it in the form $x^2 = K$.

To solve $x^2 = K$, we are required to find some number, $x$, that when squared produces $K$. This number, $x$, must be a square root of $K$. If $K$ is greater than zero, we know that it prossesses two square roots, $\sqrt{K}$ and $-\sqrt{K}$. We also know that

$(\sqrt{K})^2) = (\sqrt{K})(\sqrt{K}) = K$ and $(-\sqrt{K}) = (-\sqrt{K})(-\sqrt{K}) = K$

We now have two replacements for $x$ that produce true statements when substitued into the equation. Thus, $x = \sqrt{K}$ and $x = -\sqrt{K}$ are both solutions to $x^2 = K$. We use the notation $x = \pm \sqrt{K}$ to denote both the principal and secondary square roots.

The Nature of Solutions

Solutions of $x^2 = K$

For quadratic equations of the form $x^2 = K$,

1. If $K$ is greater than or equal to zero, the solutions are $\pm \sqrt{K}$.
2. If $K$ is negative, no real number solutions exist.
3. If $K$ is zero, the only solution is $0$.

Sample Set A

Solve each of the following quadratic equations using the method of extraction of roots.

Example $\PageIndex{1}$

$\begin{array}{flushleft} x^2 - 49 &= 0 & \text{ Rewrite }\\ x^2 &= 49\\ x &= \pm \sqrt{49}\\ x &= \pm 7 \end{array}$

Check:

$\begin{array}{flushleft} (7)^2 = 49 & \text{ Is this correct? } & (-7)^2 = 49 & \text{ Is this correct? }\\ 49 = 49 & \text{ Yes, this is correct. } & 49 = 49 & \text{ Yes, this is correct. } \end{array}$

Example $\PageIndex{2}$

$\begin{array}{flushleft} 25a^2 &= 36\\ a^2 &= \dfrac{36}{25}\\ a &= \pm \sqrt{\frac{36}{25}}\\ a &= \pm \dfrac{6}{5} \end{array}$

Check:

$\begin{array}{flushleft} 25(\dfrac{6}{5})^2 &= 36 & \text{ Is this correct? } & 25(\dfrac{-6}{5})^2 &= 36 & \text{ Is this correct? }\\ 25(\dfrac{36}{25})^2 &= 36 & \text{ Is this correct? } & 25(\dfrac{36}{25}) &= 36 & \text{ Is this correct? }\\ 36 &= 36 & \text{Yes, this is correct. } & 36 &= 36 & \text{ Yes, this is correct. } \end{array}$

Example $\PageIndex{3}$

$\begin{array}{flushleft} 4m^2 - 32 &= 0\\ 4m^2 &= 32\\ m^2 &= \dfrac{32}{4}\\ m^2 &= 8\\ m &= \pm \sqrt{8}\\ m &= \pm 2\sqrt{2} \end{array}$

Check:

$\begin{array}{flushleft} 4(2\sqrt{2})^2 &= 32 & \text{ Is this correct? } & 4(-2\sqrt{2})^2 &= 32 & \text{ Is this correct? }\\ 4[2^2(\sqrt{2})^2] &= 32 & \text{ Is this correct? } & 4[(-2)^2(\sqrt{2})^2] &= 32 & \text{ Is this correct? }\\ 4[4 \cdot 2] &= 32 & \text{ Is this correct? } & 4[4 \cdot 2] &= 32 & \text{ Is this correct? }\\ 4 \cdot 8 &= 32 & \text{ Is this correct? } & 4 \cdot 8 &= 32 & \text{ Is this correct? }\\ 32 &= 32 & \text{ Yes, this is correct. } & 32 &=32 & \text{ Yes, this is correct. } \end{array}$

Example $\PageIndex{4}$

Solve $5x^2 - 15y^2z^7 = 0$ for $x$.

$\begin{array}{flushleft} 5x^2 &= 15y^2z^7 & \text{ Divide both sides by } 5\\ x^2 &= 3y^2z^7\\ x &= \pm \sqrt{3y^2z^7}\\ x &= \pm yz^3\sqrt{3z} \end{array}$

Example $\PageIndex{5}$

Calculator Problem:

Solve $14a^2 - 235 = 0$. Round to the nearest hundredth.

$\begin{array}{flushleft} 14a^2 - 235 &= 0 & \text{ Rewrite }\\ 14a^2 &= 235 & \text{ Divide both sides by } 14\\ a^2 &= \dfrac{235}{14} \end{array}$

Rounding to the nearest hundredth produces $4.10$. We must be sure to insert the $\pm$ symbol.

$a \approx \pm 4.10$.

Example $\PageIndex{6}$

$\begin{array}{flushleft} k^2 &= -64\\ k &= \pm \sqrt{-64} \end{array}$

The radicand is negative so no real number solutions exist

Practice Set A

Solve each of the following quadratic equations using the method of extraction of roots.

Practice Problem $\PageIndex{1}$

$x^2 - 144 = 0$

Answer

$x = \pm 12$

Practice Problem $\PageIndex{2}$

$9y^2 - 121 = 0$

Answer

$y = \pm \dfrac{11}{3}$

Practice Problem $\PageIndex{3}$

$6a^2 = 108$

Answer

$a = \pm 3\sqrt{2}$

Practice Problem $\PageIndex{4}$

Solve $4n^2 = 24m^2p^8$ for $n$.

Answer

$n = \pm mp^4\sqrt{6}$

Practice Problem $\PageIndex{5}$

Solve $5p^2q^2 = 45p^2$ for $q$

Answer

$q = \pm 3$

Practice Problem $\PageIndex{6}$

Solve $16m^2 - 2206 = 0$. Round to the nearest hundredth.

Answer

$m = \pm 11.74$

Practice Problem $\PageIndex{7}$

$h^2 = -100$

Sample Set B

Solve each of the following quadratic equations using the method of extraction of roots.

Example $\PageIndex{7}$

$\begin{array}{flushleft} (x+2)^2 &= 81\\ x + 2 &= \pm \sqrt{81}\\ x + 2 &= \pm 9 & \text{ Subtract } 2 \text{ from both sides.}\\ x &= -2 \pm 9\\ x &= -2 + 9 & \text{and} & x&= -2 - 9\\ x &= 7 & & x &= -11 \end{array}$

Example $\PageIndex{8}$

$\begin{array}{flushleft} (a+3)^2 &= 5\\ a + 3 &= \pm \sqrt{5} & \text{ Subtract } 3 \text{ from both sides }\\ a &= -3 \pm \sqrt{5} \end{array}$

Practice Set B

Solve each of the following quadratic equations using the method of extraction of roots.

Practice Problem $\PageIndex{8}$

$(a + 6)^2 = 64$

Answer

$a=2,−14$

Practice Problem $\PageIndex{9}$

$(m - 4)^2 = 15$

Answer

$m = 4 \pm \sqrt{15}$

Practice Problem $\PageIndex{10}$

$(y-7)^2 = 49$

Answer

$y=0, 14$

Practice Problem $\PageIndex{11}$

$(k - 1)^2 = 12$

Answer

$k = 1 \pm 2 \sqrt{3}$

Practice Problem $\PageIndex{12}$

$(x - 11)^2 = 0$

Answer

$x=11$

Exercises

For the following problems, solve each of the quadratic equations using the method of extraction of roots.

Exercise $\PageIndex{1}$

$x^2 = 36$

Answer

$x = \pm 6$

Exercise $\PageIndex{2}$

$x^2 = 49$

Exercise $\PageIndex{3}$

$a^2 = 9$

Answer

$a = \pm 3$

Exercise $\PageIndex{4}$

$a^2 = 4$

Exercise $\PageIndex{5}$

$b^2 = 1$

Answer

$b = \pm 1$

Exercise $\PageIndex{6}$

$a^2 = 1$

Exercise $\PageIndex{7}$

$x^2 = 25$

Answer

$x = \pm 5$

Exercise $\PageIndex{8}$

$x^2 = 81$

Exercise $\PageIndex{9}$

$a^2 = 5$

Answer

$a = \pm \sqrt{5}$

Exercise $\PageIndex{10}$

$a^2 = 10$

Exercise $\PageIndex{11}$

$b^2 = 12$

Answer

$b = \pm 2\sqrt{3}$

Exercise $\PageIndex{12}$

$b^2 = 6$

Exercise $\PageIndex{13}$

$y^2 = 3$

Answer

$y = \pm \sqrt{3}$

Exercise $\PageIndex{14}$

$y^2 = 7$

Exercise $\PageIndex{15}$

$a^2 - 8 = 0$

Answer

$a = \pm 2\sqrt{2}$

Exercise $\PageIndex{16}$

$a^2 - 3 = 0$

Exercise $\PageIndex{17}$

$a^2 - 5 = 0$

Answer

$a = \pm \sqrt{5}$

Exercise $\PageIndex{18}$

$y^2 - 1 = 0$

Exercise $\PageIndex{19}$

$x^2 - 10 = 0$

Answer

$x = \pm \sqrt{10}$

Exercise $\PageIndex{20}$

$x^2 - 11 = 0$

Exercise $\PageIndex{21}$

$3x^2 - 27 = 0$

Answer

$x = \pm 3$

Exercise $\PageIndex{22}$

$5b^2 - 5 = 0$

Exercise $\PageIndex{23}$

$2x^2 = 50$

Answer

$x = \pm 5$

Exercise $\PageIndex{24}$

$4a^2 = 40$

Exercise $\PageIndex{25}$

$2x^2 = 24$

Answer

$x = \pm 2\sqrt{3}$

For the following problems, solve for the indicated variable.

Exercise $\PageIndex{26}$

$x^2 = 4a^2$, for $x$

Exercise $\PageIndex{27}$

$x^2 = 9b^2$, for $x$

Answer

$x = \pm 3b$

Exercise $\PageIndex{28}$

$a^2 = 25c^2$, for $a$

Exercise $\PageIndex{29}$

$k^2 = m^2n^2$, for $k$.

Answer

$k = \pm mn$

Exercise $\PageIndex{30}$

$k^2 = p^2q^2r^2$, for $k$

Exercise $\PageIndex{31}$

$2y^2 = 2a^2n^2$, for $y$

Answer

$y = \pm an$

Exercise $\PageIndex{32}$

$9y^2 = 27x^2z^4$, for $y$.

Exercise $\PageIndex{33}$

$x^2 - z^2 = 0$, for $x$

Answer

$x = \pm z$

Exercise $\PageIndex{34}$

$x^2 - z^2 = 0$, for $z$

Exercise $\PageIndex{35}$

$5a^2 - 10b^2 = 0$, for $a$

Answer

$a = b\sqrt{2}, -b\sqrt{2}$

For the following problems, solve each of the quadratic equations using the method of extraction of roots.

Exercise $\PageIndex{36}$

$(x-1)^2 = 4$

Exercise $\PageIndex{37}$

$(x-2)^2 = 9$

Answer

$x=5,−1$

Exercise $\PageIndex{38}$

$(x-3)^2 = 25$

Exercise $\PageIndex{39}$

$(a-5)^2 = 36$

Answer

$x=11,−1$

Exercise $\PageIndex{40}$

$(a + 3)^2 = 49$

Exercise $\PageIndex{41}$

$(a + 9)^2 = 1$

Answer

$a=−8 ,−10$

Exercise $\PageIndex{42}$

$(a - 6)^2 = 3$

Exercise $\PageIndex{43}$

$(x + 4)^2 = 5$

Answer

$a = -4 \pm \sqrt{5}$

Exercise $\PageIndex{44}$

$(b + 6)^2 = 7$

Exercise $\PageIndex{45}$

$(x + 1)^2 = a$, for $x$.

Answer

$x = -1 \pm \sqrt{a}$

Exercise $\PageIndex{46}$

$(y + 5)^2 = b$, for $y$

Exercise $\PageIndex{47}$

$(y + 2)^2 = a^2$, for $y$

Answer

$y = -2 \pm a$

Exercise $\PageIndex{48}$

$(x + 10)^2 = c^2$, for $x$

Exercise $\PageIndex{49}$

$(x - a)^2 = b^2$, for $x$

Answer

$x = a \pm b$

Exercise $\PageIndex{50}$

$(x + c)^2 = a^2$, for $x$

Calculator Problems

For the following problems, round each result to the nearest hundredth.

Exercise $\PageIndex{51}$

$8a^2 - 168 = 0$

Answer

$a = \pm 4.58$

Exercise $\PageIndex{52}$

$6m^2 - 5 = 0$

Exercise $\PageIndex{53}$

$0.03y^2 = 1.6$

Answer

$y = \pm 7.30$

Exercise $\PageIndex{54}$

$0.048x^2 = 2.01$

Exercise $\PageIndex{55}$

$1.001x^2 - 0.999 = 0$

Answer

$x = \pm 1.00$

Exercises For Review

Exercise $\PageIndex{56}$

Graph the linear inequality $3(x + 2) < 2(3x + 4)$

Exercise $\PageIndex{57}$

Solve the fractional equation: $\dfrac{x-1}{x + 4} = \dfrac{x + 3}{x - 1}$

Answer

$x = \dfrac{-11}{9}$

Exercise $\PageIndex{58}$

Find the product: $\sqrt{32x^3y^5} \sqrt{2x^3y^3}$

Exercise $\PageIndex{59}$

Solve $x^2 - 4x = 0$

Answer

$x = 0, 4$

Exercise $\PageIndex{60}$

Solve $y^2 - 8y = -12$