Solve 5g-7^2+g^2

$$5 g - 7 ^ { 2 } + g ^ { 2 }$$

Factor

$$\left(g-\frac{-\sqrt{221}-5}{2}\right)\left(g-\frac{\sqrt{221}-5}{2}\right)$$

Steps Using the Quadratic Formula

Calculate 7 to the power of 2 and get 49.

$$factor(5g-49+g^{2})$$

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.

$$g^{2}+5g-49=0$$

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.

$$g=\frac{-5±\sqrt{5^{2}-4\left(-49\right)}}{2}$$

Square 5.

$$g=\frac{-5±\sqrt{25-4\left(-49\right)}}{2}$$

Multiply -4 times -49.

$$g=\frac{-5±\sqrt{25+196}}{2}$$

Add 25 to 196.

$$g=\frac{-5±\sqrt{221}}{2}$$

Now solve the equation g=\frac{-5±\sqrt{221}}{2} when ± is plus. Add -5 to \sqrt{221}.

$$g=\frac{\sqrt{221}-5}{2}$$

Now solve the equation g=\frac{-5±\sqrt{221}}{2} when ± is minus. Subtract \sqrt{221} from -5.

$$g=\frac{-\sqrt{221}-5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{221}}{2} for x_{1} and \frac{-5-\sqrt{221}}{2} for x_{2}.

$$g^{2}+5g-49=\left(g-\frac{\sqrt{221}-5}{2}\right)\left(g-\frac{-\sqrt{221}-5}{2}\right)$$

Evaluate

$$g^{2}+5g-49$$

Solution Steps

Calculate 7 to the power of 2 and get 49.

$$5g-49+g^{2}$$