Solve (x+10)(2x-3)=54

$$(x+10)(2x-3)=54$$

Solve for x

$$x=-12$$

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+10 by 2x-3 and combine like terms.

$$2x^{2}+17x-30=54$$

Subtract 54 from both sides.

$$2x^{2}+17x-30-54=0$$

Subtract 54 from -30 to get -84.

$$2x^{2}+17x-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 17 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 2\left(-84\right)}}{2\times 2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 2\left(-84\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-17±\sqrt{289-8\left(-84\right)}}{2\times 2}$$

Multiply -8 times -84.

$$x=\frac{-17±\sqrt{289+672}}{2\times 2}$$

Add 289 to 672.

$$x=\frac{-17±\sqrt{961}}{2\times 2}$$

Take the square root of 961.

$$x=\frac{-17±31}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-17±31}{4}$$

Now solve the equation x=\frac{-17±31}{4} when ± is plus. Add -17 to 31.

$$x=\frac{14}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{2}$$

Now solve the equation x=\frac{-17±31}{4} when ± is minus. Subtract 31 from -17.

$$x=-\frac{48}{4}$$

Divide -48 by 4.

$$x=-12$$

The equation is now solved.

$$x=\frac{7}{2} x=-12$$

Steps for Completing the Square

Use the distributive property to multiply x+10 by 2x-3 and combine like terms.

$$2x^{2}+17x-30=54$$

Add 30 to both sides.

$$2x^{2}+17x=54+30$$

Add 54 and 30 to get 84.

$$2x^{2}+17x=84$$

Divide both sides by 2.

$$\frac{2x^{2}+17x}{2}=\frac{84}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{17}{2}x=\frac{84}{2}$$

Divide 84 by 2.

$$x^{2}+\frac{17}{2}x=42$$

Divide \frac{17}{2}, the coefficient of the x term, by 2 to get \frac{17}{4}. Then add the square of \frac{17}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17}{2}x+\left(\frac{17}{4}\right)^{2}=42+\left(\frac{17}{4}\right)^{2}$$

Square \frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=42+\frac{289}{16}$$

Add 42 to \frac{289}{16}.

$$x^{2}+\frac{17}{2}x+\frac{289}{16}=\frac{961}{16}$$

Factor x^{2}+\frac{17}{2}x+\frac{289}{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}.

$$\left(x+\frac{17}{4}\right)^{2}=\frac{961}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{4}\right)^{2}}=\sqrt{\frac{961}{16}}$$

Simplify.

$$x+\frac{17}{4}=\frac{31}{4} x+\frac{17}{4}=-\frac{31}{4}$$

Subtract \frac{17}{4} from both sides of the equation.

$$x=\frac{7}{2} x=-12$$