Solve left(2x-1right)^2-left(x-3right)left(5x-2right)=0

$${ \left( 2x-1 \right) }^{ 2 } - \left( x-3 \right) \left( 5x-2 \right) = 0$$

Solve for x

$$x = \frac{\sqrt{149} + 13}{2} \approx 12.603277808$$

$$x=\frac{13-\sqrt{149}}{2}\approx 0.396722192$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1-\left(x-3\right)\left(5x-2\right)=0$$

Use the distributive property to multiply x-3 by 5x-2 and combine like terms.

$$4x^{2}-4x+1-\left(5x^{2}-17x+6\right)=0$$

To find the opposite of 5x^{2}-17x+6, find the opposite of each term.

$$4x^{2}-4x+1-5x^{2}+17x-6=0$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}-4x+1+17x-6=0$$

Combine -4x and 17x to get 13x.

$$-x^{2}+13x+1-6=0$$

Subtract 6 from 1 to get -5.

$$-x^{2}+13x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 13 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-13±\sqrt{169+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$x=\frac{-13±\sqrt{169-20}}{2\left(-1\right)}$$

Add 169 to -20.

$$x=\frac{-13±\sqrt{149}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-13±\sqrt{149}}{-2}$$

Now solve the equation x=\frac{-13±\sqrt{149}}{-2} when ± is plus. Add -13 to \sqrt{149}.

$$x=\frac{\sqrt{149}-13}{-2}$$

Divide -13+\sqrt{149} by -2.

$$x=\frac{13-\sqrt{149}}{2}$$

Now solve the equation x=\frac{-13±\sqrt{149}}{-2} when ± is minus. Subtract \sqrt{149} from -13.

$$x=\frac{-\sqrt{149}-13}{-2}$$

Divide -13-\sqrt{149} by -2.

$$x=\frac{\sqrt{149}+13}{2}$$

The equation is now solved.

$$x=\frac{13-\sqrt{149}}{2} x=\frac{\sqrt{149}+13}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1-\left(x-3\right)\left(5x-2\right)=0$$

Use the distributive property to multiply x-3 by 5x-2 and combine like terms.

$$4x^{2}-4x+1-\left(5x^{2}-17x+6\right)=0$$

To find the opposite of 5x^{2}-17x+6, find the opposite of each term.

$$4x^{2}-4x+1-5x^{2}+17x-6=0$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}-4x+1+17x-6=0$$

Combine -4x and 17x to get 13x.

$$-x^{2}+13x+1-6=0$$

Subtract 6 from 1 to get -5.

$$-x^{2}+13x-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-x^{2}+13x=5$$

Divide both sides by -1.

$$\frac{-x^{2}+13x}{-1}=\frac{5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{13}{-1}x=\frac{5}{-1}$$

Divide 13 by -1.

$$x^{2}-13x=\frac{5}{-1}$$

Divide 5 by -1.

$$x^{2}-13x=-5$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-5+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-5+\frac{169}{4}$$

Add -5 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{149}{4}$$

Factor x^{2}-13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{149}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{149}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{\sqrt{149}}{2} x-\frac{13}{2}=-\frac{\sqrt{149}}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{\sqrt{149}+13}{2} x=\frac{13-\sqrt{149}}{2}$$