Solve x^2+2x-3=-3/4x+3/4

$${ x }^{ 2 } +2x-3 = - \frac{ 3 }{ 4 } x+ \frac{ 3 }{ 4 }$$

Solve for x

$$x = -\frac{15}{4} = -3\frac{3}{4} = -3.75$$

$$x=1$$

Steps Using the Quadratic Formula

Add \frac{3}{4}x to both sides.

$$x^{2}+2x-3+\frac{3}{4}x=\frac{3}{4}$$

Combine 2x and \frac{3}{4}x to get \frac{11}{4}x.

$$x^{2}+\frac{11}{4}x-3=\frac{3}{4}$$

Subtract \frac{3}{4} from both sides.

$$x^{2}+\frac{11}{4}x-3-\frac{3}{4}=0$$

Subtract \frac{3}{4} from -3 to get -\frac{15}{4}.

$$x^{2}+\frac{11}{4}x-\frac{15}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{11}{4} for b, and -\frac{15}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{11}{4}±\sqrt{\left(\frac{11}{4}\right)^{2}-4\left(-\frac{15}{4}\right)}}{2}$$

Square \frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{11}{4}±\sqrt{\frac{121}{16}-4\left(-\frac{15}{4}\right)}}{2}$$

Multiply -4 times -\frac{15}{4}.

$$x=\frac{-\frac{11}{4}±\sqrt{\frac{121}{16}+15}}{2}$$

Add \frac{121}{16} to 15.

$$x=\frac{-\frac{11}{4}±\sqrt{\frac{361}{16}}}{2}$$

Take the square root of \frac{361}{16}.

$$x=\frac{-\frac{11}{4}±\frac{19}{4}}{2}$$

Now solve the equation x=\frac{-\frac{11}{4}±\frac{19}{4}}{2} when ± is plus. Add -\frac{11}{4} to \frac{19}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-\frac{11}{4}±\frac{19}{4}}{2} when ± is minus. Subtract \frac{19}{4} from -\frac{11}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{15}{2}}{2}$$

Divide -\frac{15}{2} by 2.

$$x=-\frac{15}{4}$$

The equation is now solved.

$$x=1 x=-\frac{15}{4}$$

Steps for Completing the Square

Add \frac{3}{4}x to both sides.

$$x^{2}+2x-3+\frac{3}{4}x=\frac{3}{4}$$

Combine 2x and \frac{3}{4}x to get \frac{11}{4}x.

$$x^{2}+\frac{11}{4}x-3=\frac{3}{4}$$

Add 3 to both sides.

$$x^{2}+\frac{11}{4}x=\frac{3}{4}+3$$

Add \frac{3}{4} and 3 to get \frac{15}{4}.

$$x^{2}+\frac{11}{4}x=\frac{15}{4}$$

Divide \frac{11}{4}, the coefficient of the x term, by 2 to get \frac{11}{8}. Then add the square of \frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{4}x+\left(\frac{11}{8}\right)^{2}=\frac{15}{4}+\left(\frac{11}{8}\right)^{2}$$

Square \frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{15}{4}+\frac{121}{64}$$

Add \frac{15}{4} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{361}{64}$$

Factor x^{2}+\frac{11}{4}x+\frac{121}{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}.

$$\left(x+\frac{11}{8}\right)^{2}=\frac{361}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{8}\right)^{2}}=\sqrt{\frac{361}{64}}$$

Simplify.

$$x+\frac{11}{8}=\frac{19}{8} x+\frac{11}{8}=-\frac{19}{8}$$

Subtract \frac{11}{8} from both sides of the equation.

$$x=1 x=-\frac{15}{4}$$