Solve (2x-3)(-2x+6)=-x+3

$$(2x-3)(-2x+6)=-x+3$$

Solve for x

$$x=3$$

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-3 by -2x+6 and combine like terms.

$$-4x^{2}+18x-18=-x+3$$

Add x to both sides.

$$-4x^{2}+18x-18+x=3$$

Combine 18x and x to get 19x.

$$-4x^{2}+19x-18=3$$

Subtract 3 from both sides.

$$-4x^{2}+19x-18-3=0$$

Subtract 3 from -18 to get -21.

$$-4x^{2}+19x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 19 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-4\right)\left(-21\right)}}{2\left(-4\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-4\right)\left(-21\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-19±\sqrt{361+16\left(-21\right)}}{2\left(-4\right)}$$

Multiply 16 times -21.

$$x=\frac{-19±\sqrt{361-336}}{2\left(-4\right)}$$

Add 361 to -336.

$$x=\frac{-19±\sqrt{25}}{2\left(-4\right)}$$

Take the square root of 25.

$$x=\frac{-19±5}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-19±5}{-8}$$

Now solve the equation x=\frac{-19±5}{-8} when ± is plus. Add -19 to 5.

$$x=-\frac{14}{-8}$$

Reduce the fraction \frac{-14}{-8} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{4}$$

Now solve the equation x=\frac{-19±5}{-8} when ± is minus. Subtract 5 from -19.

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

Divide -24 by -8.

$$x=3$$

The equation is now solved.

$$x=\frac{7}{4} x=3$$

Steps for Completing the Square

Use the distributive property to multiply 2x-3 by -2x+6 and combine like terms.

$$-4x^{2}+18x-18=-x+3$$

Add x to both sides.

$$-4x^{2}+18x-18+x=3$$

Combine 18x and x to get 19x.

$$-4x^{2}+19x-18=3$$

Add 18 to both sides.

$$-4x^{2}+19x=3+18$$

Add 3 and 18 to get 21.

$$-4x^{2}+19x=21$$

Divide both sides by -4.

$$\frac{-4x^{2}+19x}{-4}=\frac{21}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{19}{-4}x=\frac{21}{-4}$$

Divide 19 by -4.

$$x^{2}-\frac{19}{4}x=\frac{21}{-4}$$

Divide 21 by -4.

$$x^{2}-\frac{19}{4}x=-\frac{21}{4}$$

Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{4}x+\left(-\frac{19}{8}\right)^{2}=-\frac{21}{4}+\left(-\frac{19}{8}\right)^{2}$$

Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=-\frac{21}{4}+\frac{361}{64}$$

Add -\frac{21}{4} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=\frac{25}{64}$$

Factor x^{2}-\frac{19}{4}x+\frac{361}{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}.

$$\left(x-\frac{19}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x-\frac{19}{8}=\frac{5}{8} x-\frac{19}{8}=-\frac{5}{8}$$

Add \frac{19}{8} to both sides of the equation.

$$x=3 x=\frac{7}{4}$$