Solve left(4xright)^2-7x-15=0

$${ \left(4x \right) }^{ 2 } -7x-15 = 0$$

Solve for x

$$x = \frac{\sqrt{1009} + 7}{32} \approx 1.211398761$$

$$x=\frac{7-\sqrt{1009}}{32}\approx -0.773898761$$

Steps Using the Quadratic Formula

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-7x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -7 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -7.

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

Multiply -4 times 16.

$$x=\frac{-\left(-7\right)±\sqrt{49-64\left(-15\right)}}{2\times 16}$$

Multiply -64 times -15.

$$x=\frac{-\left(-7\right)±\sqrt{49+960}}{2\times 16}$$

Add 49 to 960.

$$x=\frac{-\left(-7\right)±\sqrt{1009}}{2\times 16}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{1009}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{7±\sqrt{1009}}{32}$$

Now solve the equation x=\frac{7±\sqrt{1009}}{32} when ± is plus. Add 7 to \sqrt{1009}.

$$x=\frac{\sqrt{1009}+7}{32}$$

Now solve the equation x=\frac{7±\sqrt{1009}}{32} when ± is minus. Subtract \sqrt{1009} from 7.

$$x=\frac{7-\sqrt{1009}}{32}$$

The equation is now solved.

$$x=\frac{\sqrt{1009}+7}{32} x=\frac{7-\sqrt{1009}}{32}$$

Steps for Completing the Square

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-7x-15=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$16x^{2}-7x=15$$

Divide both sides by 16.

$$\frac{16x^{2}-7x}{16}=\frac{15}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}-\frac{7}{16}x=\frac{15}{16}$$

Divide -\frac{7}{16}, the coefficient of the x term, by 2 to get -\frac{7}{32}. Then add the square of -\frac{7}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{16}x+\left(-\frac{7}{32}\right)^{2}=\frac{15}{16}+\left(-\frac{7}{32}\right)^{2}$$

Square -\frac{7}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{16}x+\frac{49}{1024}=\frac{15}{16}+\frac{49}{1024}$$

Add \frac{15}{16} to \frac{49}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{16}x+\frac{49}{1024}=\frac{1009}{1024}$$

Factor x^{2}-\frac{7}{16}x+\frac{49}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{32}\right)^{2}=\frac{1009}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{32}\right)^{2}}=\sqrt{\frac{1009}{1024}}$$

Simplify.

$$x-\frac{7}{32}=\frac{\sqrt{1009}}{32} x-\frac{7}{32}=-\frac{\sqrt{1009}}{32}$$

Add \frac{7}{32} to both sides of the equation.

$$x=\frac{\sqrt{1009}+7}{32} x=\frac{7-\sqrt{1009}}{32}$$