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Elementary Algebra (Ellis and Burzynski)
10: Quadratic Equations
10.4: Solving Quadratic Equations Using the Method of Extraction of Roots

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

Last updated

Jun 4, 2023
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- 10.3: Solving Quadratic Equations by Factoring
- 10.5: Solving Quadratic Equations Using the Method of Completing the Square

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The Method Of Extraction Of Roots

Extraction of Roots

Quadratic equations of the form can be solved by the method of extraction of roots by rewriting it in the form .

To solve , we are required to find some number, , that when squared produces . This number, , must be a square root of . If is greater than zero, we know that it prossesses two square roots, and . We also know that

and

We now have two replacements for that produce true statements when substitued into the equation. Thus, and are both solutions to . We use the notation to denote both the principal and secondary square roots.

The Nature of Solutions

Solutions of

For quadratic equations of the form ,

If is greater than or equal to zero, the solutions are .
If is negative, no real number solutions exist.
If is zero, the only solution is .

Sample Set A

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

Example

$Illegal pream-token (f)\begin{array}{flushleft} x^2 - 49 &= 0 & \text{ Rewrite }\\ x^2 &= 49\\ x &= \pm \sqrt{49}\\ x &= \pm 7 \end{array}$

Check:

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

Example

$Illegal pream-token (f)\begin{array}{flushleft} 25a^2 &= 36\\ a^2 &= \dfrac{36}{25}\\ a &= \pm \sqrt{\frac{36}{25}}\\ a &= \pm \dfrac{6}{5} \end{array}$

Check:

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

Example

$Illegal pream-token (f)\begin{array}{flushleft} 4m^2 - 32 &= 0\\ 4m^2 &= 32\\ m^2 &= \dfrac{32}{4}\\ m^2 &= 8\\ m &= \pm \sqrt{8}\\ m &= \pm 2\sqrt{2} \end{array}$

Check:

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

Example

Solve for .

$Illegal pream-token (f)\begin{array}{flushleft} 5x^2 &= 15y^2z^7 & \text{ Divide both sides by } 5\\ x^2 &= 3y^2z^7\\ x &= \pm \sqrt{3y^2z^7}\\ x &= \pm yz^3\sqrt{3z} \end{array}$

Example

Calculator Problem:

Solve . Round to the nearest hundredth.

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

Rounding to the nearest hundredth produces . We must be sure to insert the symbol.

Example

$Illegal pream-token (f)\begin{array}{flushleft} k^2 &= -64\\ k &= \pm \sqrt{-64} \end{array}$

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

Answer

Practice Problem

Answer

Practice Problem

Answer

Practice Problem

Solve for .

Answer

Practice Problem

Solve for

Answer

Practice Problem

Solve . Round to the nearest hundredth.

Answer

Practice Problem

Sample Set B

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

Example

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

Example

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

Practice Set B

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

Practice Problem

Answer

Practice Problem

Answer

Practice Problem

Answer

Practice Problem

Answer

Practice Problem

Answer

Exercises

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

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

For the following problems, solve for the indicated variable.

Exercise

, for

Exercise

, for

Answer

Exercise

, for

Exercise

, for .

Answer

Exercise

, for

Exercise

, for

Answer

Exercise

, for .

Exercise

, for

Answer

Exercise

, for

Exercise

, for

Answer

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

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercise

, for .

Answer

Exercise

, for

Exercise

, for

Answer

Exercise

, for

Exercise

, for

Answer

Exercise

, for

Calculator Problems

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

Exercise

Answer

Exercise

Answer

Exercise

Answer

Exercises For Review

Exercise

Graph the linear inequality

$A horizontal line with arrows on both ends.$

Exercise

Solve the fractional equation:

Answer

Exercise

Find the product:

Exercise

Solve

Answer

Exercise

Solve