Solve x*(4x+7)=15

$$x \cdot ( 4 x + 7 ) = 15$$

Solve for x

$$x=-3$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 4x+7.

$$4x^{2}+7x=15$$

Subtract 15 from both sides.

$$4x^{2}+7x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 7 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 4\left(-15\right)}}{2\times 4}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 4\left(-15\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-7±\sqrt{49-16\left(-15\right)}}{2\times 4}$$

Multiply -16 times -15.

$$x=\frac{-7±\sqrt{49+240}}{2\times 4}$$

Add 49 to 240.

$$x=\frac{-7±\sqrt{289}}{2\times 4}$$

Take the square root of 289.

$$x=\frac{-7±17}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-7±17}{8}$$

Now solve the equation x=\frac{-7±17}{8} when ± is plus. Add -7 to 17.

$$x=\frac{10}{8}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{5}{4}$$

Now solve the equation x=\frac{-7±17}{8} when ± is minus. Subtract 17 from -7.

$$x=-\frac{24}{8}$$

Divide -24 by 8.

$$x=-3$$

The equation is now solved.

$$x=\frac{5}{4} x=-3$$

Steps for Completing the Square

Use the distributive property to multiply x by 4x+7.

$$4x^{2}+7x=15$$

Divide both sides by 4.

$$\frac{4x^{2}+7x}{4}=\frac{15}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{7}{4}x=\frac{15}{4}$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\frac{15}{4}+\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{15}{4}+\frac{49}{64}$$

Add \frac{15}{4} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{289}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{289}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{289}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{17}{8} x+\frac{7}{8}=-\frac{17}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=\frac{5}{4} x=-3$$