Solve (2x+10)(2x+5)=100

$$( 2 x + 10 ) ( 2 x + 5 ) = 100$$

Solve for x

$$x = \frac{5 \sqrt{17} - 15}{4} \approx 1.403882032$$

$$x=\frac{-5\sqrt{17}-15}{4}\approx -8.903882032$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+10 by 2x+5 and combine like terms.

$$4x^{2}+30x+50=100$$

Subtract 100 from both sides.

$$4x^{2}+30x+50-100=0$$

Subtract 100 from 50 to get -50.

$$4x^{2}+30x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 30 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 4\left(-50\right)}}{2\times 4}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 4\left(-50\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-30±\sqrt{900-16\left(-50\right)}}{2\times 4}$$

Multiply -16 times -50.

$$x=\frac{-30±\sqrt{900+800}}{2\times 4}$$

Add 900 to 800.

$$x=\frac{-30±\sqrt{1700}}{2\times 4}$$

Take the square root of 1700.

$$x=\frac{-30±10\sqrt{17}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-30±10\sqrt{17}}{8}$$

Now solve the equation x=\frac{-30±10\sqrt{17}}{8} when ± is plus. Add -30 to 10\sqrt{17}.

$$x=\frac{10\sqrt{17}-30}{8}$$

Divide -30+10\sqrt{17} by 8.

$$x=\frac{5\sqrt{17}-15}{4}$$

Now solve the equation x=\frac{-30±10\sqrt{17}}{8} when ± is minus. Subtract 10\sqrt{17} from -30.

$$x=\frac{-10\sqrt{17}-30}{8}$$

Divide -30-10\sqrt{17} by 8.

$$x=\frac{-5\sqrt{17}-15}{4}$$

The equation is now solved.

$$x=\frac{5\sqrt{17}-15}{4} x=\frac{-5\sqrt{17}-15}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2x+10 by 2x+5 and combine like terms.

$$4x^{2}+30x+50=100$$

Subtract 50 from both sides.

$$4x^{2}+30x=100-50$$

Subtract 50 from 100 to get 50.

$$4x^{2}+30x=50$$

Divide both sides by 4.

$$\frac{4x^{2}+30x}{4}=\frac{50}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{30}{4}x=\frac{50}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{15}{2}x=\frac{50}{4}$$

Reduce the fraction \frac{50}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{15}{2}x=\frac{25}{2}$$

Divide \frac{15}{2}, the coefficient of the x term, by 2 to get \frac{15}{4}. Then add the square of \frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{2}x+\left(\frac{15}{4}\right)^{2}=\frac{25}{2}+\left(\frac{15}{4}\right)^{2}$$

Square \frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=\frac{25}{2}+\frac{225}{16}$$

Add \frac{25}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=\frac{425}{16}$$

Factor x^{2}+\frac{15}{2}x+\frac{225}{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}.

$$\left(x+\frac{15}{4}\right)^{2}=\frac{425}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{4}\right)^{2}}=\sqrt{\frac{425}{16}}$$

Simplify.

$$x+\frac{15}{4}=\frac{5\sqrt{17}}{4} x+\frac{15}{4}=-\frac{5\sqrt{17}}{4}$$

Subtract \frac{15}{4} from both sides of the equation.

$$x=\frac{5\sqrt{17}-15}{4} x=\frac{-5\sqrt{17}-15}{4}$$