Solve 4x(x+1)=15

$$4x(x+1)=15$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+1.

$$4x^{2}+4x=15$$

Subtract 15 from both sides.

$$4x^{2}+4x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 4 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-15\right)}}{2\times 4}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-15\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-4±\sqrt{16-16\left(-15\right)}}{2\times 4}$$

Multiply -16 times -15.

$$x=\frac{-4±\sqrt{16+240}}{2\times 4}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\times 4}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-4±16}{8}$$

Now solve the equation x=\frac{-4±16}{8} when ± is plus. Add -4 to 16.

$$x=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-4±16}{8} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{8}$$

Reduce the fraction \frac{-20}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

$$x=\frac{3}{2} x=-\frac{5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+1.

$$4x^{2}+4x=15$$

Divide both sides by 4.

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

Dividing by 4 undoes the multiplication by 4.

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

Divide 4 by 4.

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

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{15}{4}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{15+1}{4}$$

Add \frac{15}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=4$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{4}$$

Simplify.

$$x+\frac{1}{2}=2 x+\frac{1}{2}=-2$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{3}{2} x=-\frac{5}{2}$$