Solve 2k+3k^2+5k^2-7

$$2 k + 3 k ^ { 2 } + 5 k ^ { 2 } - 7$$

Factor

$$8\left(k-\frac{-\sqrt{57}-1}{8}\right)\left(k-\frac{\sqrt{57}-1}{8}\right)$$

Steps Using the Quadratic Formula

Combine 3k^{2} and 5k^{2} to get 8k^{2}.

$$factor(2k+8k^{2}-7)$$

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.

$$8k^{2}+2k-7=0$$

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.

$$k=\frac{-2±\sqrt{2^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Square 2.

$$k=\frac{-2±\sqrt{4-4\times 8\left(-7\right)}}{2\times 8}$$

Multiply -4 times 8.

$$k=\frac{-2±\sqrt{4-32\left(-7\right)}}{2\times 8}$$

Multiply -32 times -7.

$$k=\frac{-2±\sqrt{4+224}}{2\times 8}$$

Add 4 to 224.

$$k=\frac{-2±\sqrt{228}}{2\times 8}$$

Take the square root of 228.

$$k=\frac{-2±2\sqrt{57}}{2\times 8}$$

Multiply 2 times 8.

$$k=\frac{-2±2\sqrt{57}}{16}$$

Now solve the equation k=\frac{-2±2\sqrt{57}}{16} when ± is plus. Add -2 to 2\sqrt{57}.

$$k=\frac{2\sqrt{57}-2}{16}$$

Divide -2+2\sqrt{57} by 16.

$$k=\frac{\sqrt{57}-1}{8}$$

Now solve the equation k=\frac{-2±2\sqrt{57}}{16} when ± is minus. Subtract 2\sqrt{57} from -2.

$$k=\frac{-2\sqrt{57}-2}{16}$$

Divide -2-2\sqrt{57} by 16.

$$k=\frac{-\sqrt{57}-1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{57}}{8} for x_{1} and \frac{-1-\sqrt{57}}{8} for x_{2}.

$$8k^{2}+2k-7=8\left(k-\frac{\sqrt{57}-1}{8}\right)\left(k-\frac{-\sqrt{57}-1}{8}\right)$$

Evaluate

$$8k^{2}+2k-7$$

Solution Steps

Combine 3k^{2} and 5k^{2} to get 8k^{2}.

$$2k+8k^{2}-7$$