Solve (-7x-2)^2-25=0

$$( - 7 x - 2 ) ^ { 2 } - 25 = 0$$

Solve for x

$$x=-1$$

$$x=\frac{3}{7}\approx 0.428571429$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Subtract 25 from 4 to get -21.

$$49x^{2}+28x-21=0$$

Divide both sides by 7.

$$7x^{2}+4x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=7\left(-3\right)=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=-3 b=7$$

Rewrite 7x^{2}+4x-3 as \left(7x^{2}-3x\right)+\left(7x-3\right).

$$\left(7x^{2}-3x\right)+\left(7x-3\right)$$

Factor out x in 7x^{2}-3x.

$$x\left(7x-3\right)+7x-3$$

Factor out common term 7x-3 by using distributive property.

$$\left(7x-3\right)\left(x+1\right)$$

To find equation solutions, solve 7x-3=0 and x+1=0.

$$x=\frac{3}{7} x=-1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Subtract 25 from 4 to get -21.

$$49x^{2}+28x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, 28 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 49\left(-21\right)}}{2\times 49}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 49\left(-21\right)}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-28±\sqrt{784-196\left(-21\right)}}{2\times 49}$$

Multiply -196 times -21.

$$x=\frac{-28±\sqrt{784+4116}}{2\times 49}$$

Add 784 to 4116.

$$x=\frac{-28±\sqrt{4900}}{2\times 49}$$

Take the square root of 4900.

$$x=\frac{-28±70}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{-28±70}{98}$$

Now solve the equation x=\frac{-28±70}{98} when ± is plus. Add -28 to 70.

$$x=\frac{42}{98}$$

Reduce the fraction \frac{42}{98} to lowest terms by extracting and canceling out 14.

$$x=\frac{3}{7}$$

Now solve the equation x=\frac{-28±70}{98} when ± is minus. Subtract 70 from -28.

$$x=-\frac{98}{98}$$

Divide -98 by 98.

$$x=-1$$

The equation is now solved.

$$x=\frac{3}{7} x=-1$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-7x-2\right)^{2}.

$$49x^{2}+28x+4-25=0$$

Subtract 25 from 4 to get -21.

$$49x^{2}+28x-21=0$$

Add 21 to both sides. Anything plus zero gives itself.

$$49x^{2}+28x=21$$

Divide both sides by 49.

$$\frac{49x^{2}+28x}{49}=\frac{21}{49}$$

Dividing by 49 undoes the multiplication by 49.

$$x^{2}+\frac{28}{49}x=\frac{21}{49}$$

Reduce the fraction \frac{28}{49} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{4}{7}x=\frac{21}{49}$$

Reduce the fraction \frac{21}{49} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{4}{7}x=\frac{3}{7}$$

Divide \frac{4}{7}, the coefficient of the x term, by 2 to get \frac{2}{7}. Then add the square of \frac{2}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{7}x+\left(\frac{2}{7}\right)^{2}=\frac{3}{7}+\left(\frac{2}{7}\right)^{2}$$

Square \frac{2}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=\frac{3}{7}+\frac{4}{49}$$

Add \frac{3}{7} to \frac{4}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=\frac{25}{49}$$

Factor x^{2}+\frac{4}{7}x+\frac{4}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{7}\right)^{2}=\frac{25}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{7}\right)^{2}}=\sqrt{\frac{25}{49}}$$

Simplify.

$$x+\frac{2}{7}=\frac{5}{7} x+\frac{2}{7}=-\frac{5}{7}$$

Subtract \frac{2}{7} from both sides of the equation.

$$x=\frac{3}{7} x=-1$$