Solve -2x^2+5x-3

$$-2 { x }^{ 2 } +5x-3$$

Factor

$$\left(1-x\right)\left(2x-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=5 ab=-2\left(-3\right)=6$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=3 b=2$$

Rewrite -2x^{2}+5x-3 as \left(-2x^{2}+3x\right)+\left(2x-3\right).

$$\left(-2x^{2}+3x\right)+\left(2x-3\right)$$

Factor out -x in -2x^{2}+3x.

$$-x\left(2x-3\right)+2x-3$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(-x+1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-2x^{2}+5x-3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-5±\sqrt{25+8\left(-3\right)}}{2\left(-2\right)}$$

Multiply 8 times -3.

$$x=\frac{-5±\sqrt{25-24}}{2\left(-2\right)}$$

Add 25 to -24.

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

Take the square root of 1.

$$x=\frac{-5±1}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-5±1}{-4}$$

Now solve the equation x=\frac{-5±1}{-4} when ± is plus. Add -5 to 1.

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

Divide -4 by -4.

$$x=1$$

Now solve the equation x=\frac{-5±1}{-4} when ± is minus. Subtract 1 from -5.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and \frac{3}{2} for x_{2}.

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

Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Cancel out 2, the greatest common factor in -2 and 2.

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

Evaluate

$$\left(1-x\right)\left(2x-3\right)$$