Solve 3x^2-5x+1+x^2-7x-3

$$3 { x }^{ 2 } -5x+1+ { x }^{ 2 } -7x-3$$

Evaluate

$$4x^{2}-12x-2$$

Solution Steps

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-5x+1-7x-3$$

Combine -5x and -7x to get -12x.

$$4x^{2}-12x+1-3$$

Subtract 3 from 1 to get -2.

$$4x^{2}-12x-2$$

Factor

$$4\left(x-\frac{3-\sqrt{11}}{2}\right)\left(x-\frac{\sqrt{11}+3}{2}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$factor(4x^{2}-5x+1-7x-3)$$

Combine -5x and -7x to get -12x.

$$factor(4x^{2}-12x+1-3)$$

Subtract 3 from 1 to get -2.

$$factor(4x^{2}-12x-2)$$

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.

$$4x^{2}-12x-2=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-2\right)}}{2\times 4}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-2\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-2\right)}}{2\times 4}$$

Multiply -16 times -2.

$$x=\frac{-\left(-12\right)±\sqrt{144+32}}{2\times 4}$$

Add 144 to 32.

$$x=\frac{-\left(-12\right)±\sqrt{176}}{2\times 4}$$

Take the square root of 176.

$$x=\frac{-\left(-12\right)±4\sqrt{11}}{2\times 4}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{12±4\sqrt{11}}{8}$$

Now solve the equation x=\frac{12±4\sqrt{11}}{8} when ± is plus. Add 12 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}+12}{8}$$

Divide 12+4\sqrt{11} by 8.

$$x=\frac{\sqrt{11}+3}{2}$$

Now solve the equation x=\frac{12±4\sqrt{11}}{8} when ± is minus. Subtract 4\sqrt{11} from 12.

$$x=\frac{12-4\sqrt{11}}{8}$$

Divide 12-4\sqrt{11} by 8.

$$x=\frac{3-\sqrt{11}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{11}}{2} for x_{1} and \frac{3-\sqrt{11}}{2} for x_{2}.

$$4x^{2}-12x-2=4\left(x-\frac{\sqrt{11}+3}{2}\right)\left(x-\frac{3-\sqrt{11}}{2}\right)$$