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

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## 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 $$\PageIndex{1}$$

$$x^2 - 144 = 0$$

$$x = \pm 12$$

##### Practice Problem $$\PageIndex{2}$$

$$9y^2 - 121 = 0$$

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

##### Practice Problem $$\PageIndex{3}$$

$$6a^2 = 108$$

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

##### Practice Problem $$\PageIndex{4}$$

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

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

##### Practice Problem $$\PageIndex{5}$$

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

$$q = \pm 3$$

##### Practice Problem $$\PageIndex{6}$$

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

$$m = \pm 11.74$$

##### 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 $$\PageIndex{8}$$

$$(a + 6)^2 = 64$$

$$a=2,−14$$

##### Practice Problem $$\PageIndex{9}$$

$$(m - 4)^2 = 15$$

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

##### Practice Problem $$\PageIndex{10}$$

$$(y-7)^2 = 49$$

$$y=0, 14$$

##### Practice Problem $$\PageIndex{11}$$

$$(k - 1)^2 = 12$$

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

##### Practice Problem $$\PageIndex{12}$$

$$(x - 11)^2 = 0$$

$$x=11$$

## Exercises

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

##### Exercise $$\PageIndex{1}$$

$$x^2 = 36$$

$$x = \pm 6$$

##### Exercise $$\PageIndex{2}$$

$$x^2 = 49$$

##### Exercise $$\PageIndex{3}$$

$$a^2 = 9$$

$$a = \pm 3$$

##### Exercise $$\PageIndex{4}$$

$$a^2 = 4$$

##### Exercise $$\PageIndex{5}$$

$$b^2 = 1$$

$$b = \pm 1$$

##### Exercise $$\PageIndex{6}$$

$$a^2 = 1$$

##### Exercise $$\PageIndex{7}$$

$$x^2 = 25$$

$$x = \pm 5$$

##### Exercise $$\PageIndex{8}$$

$$x^2 = 81$$

##### Exercise $$\PageIndex{9}$$

$$a^2 = 5$$

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

##### Exercise $$\PageIndex{10}$$

$$a^2 = 10$$

##### Exercise $$\PageIndex{11}$$

$$b^2 = 12$$

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

##### Exercise $$\PageIndex{12}$$

$$b^2 = 6$$

##### Exercise $$\PageIndex{13}$$

$$y^2 = 3$$

$$y = \pm \sqrt{3}$$

##### Exercise $$\PageIndex{14}$$

$$y^2 = 7$$

##### Exercise $$\PageIndex{15}$$

$$a^2 - 8 = 0$$

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

##### Exercise $$\PageIndex{16}$$

$$a^2 - 3 = 0$$

##### Exercise $$\PageIndex{17}$$

$$a^2 - 5 = 0$$

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

##### Exercise $$\PageIndex{18}$$

$$y^2 - 1 = 0$$

##### Exercise $$\PageIndex{19}$$

$$x^2 - 10 = 0$$

$$x = \pm \sqrt{10}$$

##### Exercise $$\PageIndex{20}$$

$$x^2 - 11 = 0$$

##### Exercise $$\PageIndex{21}$$

$$3x^2 - 27 = 0$$

$$x = \pm 3$$

##### Exercise $$\PageIndex{22}$$

$$5b^2 - 5 = 0$$

##### Exercise $$\PageIndex{23}$$

$$2x^2 = 50$$

$$x = \pm 5$$

##### Exercise $$\PageIndex{24}$$

$$4a^2 = 40$$

##### Exercise $$\PageIndex{25}$$

$$2x^2 = 24$$

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

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 $$x$$

$$x = \pm 3b$$

##### Exercise $$\PageIndex{28}$$

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

##### Exercise $$\PageIndex{29}$$

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

$$k = \pm mn$$

##### Exercise $$\PageIndex{30}$$

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

##### Exercise $$\PageIndex{31}$$

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

$$y = \pm an$$

##### Exercise $$\PageIndex{32}$$

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

##### Exercise $$\PageIndex{33}$$

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

$$x = \pm z$$

##### Exercise $$\PageIndex{34}$$

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

##### Exercise $$\PageIndex{35}$$

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

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

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 $$\PageIndex{37}$$

$$(x-2)^2 = 9$$

$$x=5,−1$$

##### Exercise $$\PageIndex{38}$$

$$(x-3)^2 = 25$$

##### Exercise $$\PageIndex{39}$$

$$(a-5)^2 = 36$$

$$x=11,−1$$

##### Exercise $$\PageIndex{40}$$

$$(a + 3)^2 = 49$$

##### Exercise $$\PageIndex{41}$$

$$(a + 9)^2 = 1$$

$$a=−8 ,−10$$

##### Exercise $$\PageIndex{42}$$

$$(a - 6)^2 = 3$$

##### Exercise $$\PageIndex{43}$$

$$(x + 4)^2 = 5$$

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

##### Exercise $$\PageIndex{44}$$

$$(b + 6)^2 = 7$$

##### Exercise $$\PageIndex{45}$$

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

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

##### Exercise $$\PageIndex{46}$$

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

##### Exercise $$\PageIndex{47}$$

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

$$y = -2 \pm a$$

##### Exercise $$\PageIndex{48}$$

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

##### Exercise $$\PageIndex{49}$$

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

$$x = a \pm b$$

##### Exercise $$\PageIndex{50}$$

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

### Calculator Problems

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

##### Exercise $$\PageIndex{51}$$

$$8a^2 - 168 = 0$$

$$a = \pm 4.58$$

##### Exercise $$\PageIndex{52}$$

$$6m^2 - 5 = 0$$

##### Exercise $$\PageIndex{53}$$

$$0.03y^2 = 1.6$$

$$y = \pm 7.30$$

##### Exercise $$\PageIndex{54}$$

$$0.048x^2 = 2.01$$

##### Exercise $$\PageIndex{55}$$

$$1.001x^2 - 0.999 = 0$$

$$x = \pm 1.00$$

## Exercises For Review

##### Exercise $$\PageIndex{56}$$

Graph the linear inequality $$3(x + 2) < 2(3x + 4)$$ ##### Exercise $$\PageIndex{57}$$

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

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

##### Exercise $$\PageIndex{58}$$

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

##### Exercise $$\PageIndex{59}$$

Solve $$x^2 - 4x = 0$$

$$x = 0, 4$$
##### Exercise $$\PageIndex{60}$$
Solve $$y^2 - 8y = -12$$