Solve left(2x-7right)^2-7(7+2x)=0

$${ \left(2x-7 \right) }^{ 2 } -7(7+2x)=0$$

Solve for x

$$x = \frac{21}{2} = 10\frac{1}{2} = 10.5$$

$$x=0$$

Steps Using Factoring

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

$$4x^{2}-28x+49-7\left(7+2x\right)=0$$

Use the distributive property to multiply -7 by 7+2x.

$$4x^{2}-28x+49-49-14x=0$$

Subtract 49 from 49 to get 0.

$$4x^{2}-28x-14x=0$$

Combine -28x and -14x to get -42x.

$$4x^{2}-42x=0$$

Factor out x.

$$x\left(4x-42\right)=0$$

To find equation solutions, solve x=0 and 4x-42=0.

$$x=0 x=\frac{21}{2}$$

Steps Using the Quadratic Formula

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

$$4x^{2}-28x+49-7\left(7+2x\right)=0$$

Use the distributive property to multiply -7 by 7+2x.

$$4x^{2}-28x+49-49-14x=0$$

Subtract 49 from 49 to get 0.

$$4x^{2}-28x-14x=0$$

Combine -28x and -14x to get -42x.

$$4x^{2}-42x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -42 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}}}{2\times 4}$$

Take the square root of \left(-42\right)^{2}.

$$x=\frac{-\left(-42\right)±42}{2\times 4}$$

The opposite of -42 is 42.

$$x=\frac{42±42}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{42±42}{8}$$

Now solve the equation x=\frac{42±42}{8} when ± is plus. Add 42 to 42.

$$x=\frac{84}{8}$$

Reduce the fraction \frac{84}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{21}{2}$$

Now solve the equation x=\frac{42±42}{8} when ± is minus. Subtract 42 from 42.

$$x=\frac{0}{8}$$

Divide 0 by 8.

$$x=0$$

The equation is now solved.

$$x=\frac{21}{2} x=0$$

Steps for Completing the Square

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

$$4x^{2}-28x+49-7\left(7+2x\right)=0$$

Use the distributive property to multiply -7 by 7+2x.

$$4x^{2}-28x+49-49-14x=0$$

Subtract 49 from 49 to get 0.

$$4x^{2}-28x-14x=0$$

Combine -28x and -14x to get -42x.

$$4x^{2}-42x=0$$

Divide both sides by 4.

$$\frac{4x^{2}-42x}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{42}{4}\right)x=\frac{0}{4}$$

Reduce the fraction \frac{-42}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{21}{2}x=\frac{0}{4}$$

Divide 0 by 4.

$$x^{2}-\frac{21}{2}x=0$$

Divide -\frac{21}{2}, the coefficient of the x term, by 2 to get -\frac{21}{4}. Then add the square of -\frac{21}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{2}x+\left(-\frac{21}{4}\right)^{2}=\left(-\frac{21}{4}\right)^{2}$$

Square -\frac{21}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{2}x+\frac{441}{16}=\frac{441}{16}$$

Factor x^{2}-\frac{21}{2}x+\frac{441}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{4}\right)^{2}=\frac{441}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Simplify.

$$x-\frac{21}{4}=\frac{21}{4} x-\frac{21}{4}=-\frac{21}{4}$$

Add \frac{21}{4} to both sides of the equation.

$$x=\frac{21}{2} x=0$$