Solve 1428=(18+2x)(26+2x)

$$1428 = ( 18 + 2 x ) ( 26 + 2 x )$$

Solve for x

$$x=-30$$

$$x=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 18+2x by 26+2x and combine like terms.

$$1428=468+88x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$468+88x+4x^{2}=1428$$

Subtract 1428 from both sides.

$$468+88x+4x^{2}-1428=0$$

Subtract 1428 from 468 to get -960.

$$-960+88x+4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$4x^{2}+88x-960=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 88 for b, and -960 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-88±\sqrt{88^{2}-4\times 4\left(-960\right)}}{2\times 4}$$

Square 88.

$$x=\frac{-88±\sqrt{7744-4\times 4\left(-960\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-88±\sqrt{7744-16\left(-960\right)}}{2\times 4}$$

Multiply -16 times -960.

$$x=\frac{-88±\sqrt{7744+15360}}{2\times 4}$$

Add 7744 to 15360.

$$x=\frac{-88±\sqrt{23104}}{2\times 4}$$

Take the square root of 23104.

$$x=\frac{-88±152}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-88±152}{8}$$

Now solve the equation x=\frac{-88±152}{8} when ± is plus. Add -88 to 152.

$$x=\frac{64}{8}$$

Divide 64 by 8.

$$x=8$$

Now solve the equation x=\frac{-88±152}{8} when ± is minus. Subtract 152 from -88.

$$x=-\frac{240}{8}$$

Divide -240 by 8.

$$x=-30$$

The equation is now solved.

$$x=8 x=-30$$

Steps for Completing the Square

Use the distributive property to multiply 18+2x by 26+2x and combine like terms.

$$1428=468+88x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$468+88x+4x^{2}=1428$$

Subtract 468 from both sides.

$$88x+4x^{2}=1428-468$$

Subtract 468 from 1428 to get 960.

$$88x+4x^{2}=960$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}+88x=960$$

Divide both sides by 4.

$$\frac{4x^{2}+88x}{4}=\frac{960}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{88}{4}x=\frac{960}{4}$$

Divide 88 by 4.

$$x^{2}+22x=\frac{960}{4}$$

Divide 960 by 4.

$$x^{2}+22x=240$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+22x+11^{2}=240+11^{2}$$

Square 11.

$$x^{2}+22x+121=240+121$$

Add 240 to 121.

$$x^{2}+22x+121=361$$

Factor x^{2}+22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11\right)^{2}=361$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{361}$$

Simplify.

$$x+11=19 x+11=-19$$

Subtract 11 from both sides of the equation.

$$x=8 x=-30$$