Solve left(4x+3right)left(4x-1right)=21

$$\left( 4x+3 \right) \left( 4x-1 \right) = 21$$

Solve for x

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

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x+3 by 4x-1 and combine like terms.

$$16x^{2}+8x-3=21$$

Subtract 21 from both sides.

$$16x^{2}+8x-3-21=0$$

Subtract 21 from -3 to get -24.

$$16x^{2}+8x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 8 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 16\left(-24\right)}}{2\times 16}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 16\left(-24\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-8±\sqrt{64-64\left(-24\right)}}{2\times 16}$$

Multiply -64 times -24.

$$x=\frac{-8±\sqrt{64+1536}}{2\times 16}$$

Add 64 to 1536.

$$x=\frac{-8±\sqrt{1600}}{2\times 16}$$

Take the square root of 1600.

$$x=\frac{-8±40}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-8±40}{32}$$

Now solve the equation x=\frac{-8±40}{32} when ± is plus. Add -8 to 40.

$$x=\frac{32}{32}$$

Divide 32 by 32.

$$x=1$$

Now solve the equation x=\frac{-8±40}{32} when ± is minus. Subtract 40 from -8.

$$x=-\frac{48}{32}$$

Reduce the fraction \frac{-48}{32} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 4x+3 by 4x-1 and combine like terms.

$$16x^{2}+8x-3=21$$

Add 3 to both sides.

$$16x^{2}+8x=21+3$$

Add 21 and 3 to get 24.

$$16x^{2}+8x=24$$

Divide both sides by 16.

$$\frac{16x^{2}+8x}{16}=\frac{24}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{8}{16}x=\frac{24}{16}$$

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{1}{2}x=\frac{24}{16}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{1}{2}x=\frac{3}{2}$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{3}{2}+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{3}{2}+\frac{1}{16}$$

Add \frac{3}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{25}{16}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{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}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{5}{4} x+\frac{1}{4}=-\frac{5}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

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