Solve -3x^2+76x-25

$$-3 { x }^{ 2 } +76x-25$$

Factor

$$\left(25-x\right)\left(3x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx-25. To find a and b, set up a system to be solved.

$$a+b=76 ab=-3\left(-25\right)=75$$

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 75.

$$1,75 3,25 5,15$$

Calculate the sum for each pair.

$$1+75=76 3+25=28 5+15=20$$

The solution is the pair that gives sum 76.

$$a=75 b=1$$

Rewrite -3x^{2}+76x-25 as \left(-3x^{2}+75x\right)+\left(x-25\right).

$$\left(-3x^{2}+75x\right)+\left(x-25\right)$$

Factor out 3x in the first and -1 in the second group.

$$3x\left(-x+25\right)-\left(-x+25\right)$$

Factor out common term -x+25 by using distributive property.

$$\left(-x+25\right)\left(3x-1\right)$$

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.

$$-3x^{2}+76x-25=0$$

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.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-3\right)\left(-25\right)}}{2\left(-3\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-3\right)\left(-25\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-76±\sqrt{5776+12\left(-25\right)}}{2\left(-3\right)}$$

Multiply 12 times -25.

$$x=\frac{-76±\sqrt{5776-300}}{2\left(-3\right)}$$

Add 5776 to -300.

$$x=\frac{-76±\sqrt{5476}}{2\left(-3\right)}$$

Take the square root of 5476.

$$x=\frac{-76±74}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-76±74}{-6}$$

Now solve the equation x=\frac{-76±74}{-6} when ± is plus. Add -76 to 74.

$$x=-\frac{2}{-6}$$

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

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-76±74}{-6} when ± is minus. Subtract 74 from -76.

$$x=-\frac{150}{-6}$$

Divide -150 by -6.

$$x=25$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{3} for x_{1} and 25 for x_{2}.

$$-3x^{2}+76x-25=-3\left(x-\frac{1}{3}\right)\left(x-25\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-3x^{2}+76x-25=-3\times \frac{-3x+1}{-3}\left(x-25\right)$$

Cancel out 3, the greatest common factor in -3 and 3.

$$-3x^{2}+76x-25=\left(-3x+1\right)\left(x-25\right)$$

Evaluate

$$\left(25-x\right)\left(3x-1\right)$$