Solve 1/2=b^2+25-49/2*b*7

$$\frac { 1 } { 2 } = \frac { b ^ { 2 } + 25 - 49 } { 2 \times b \times 7 }$$

Solve for b

$$b = \frac{\sqrt{145} + 7}{2} \approx 9.520797289$$

$$b=\frac{7-\sqrt{145}}{2}\approx -2.520797289$$

Steps Using the Quadratic Formula

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 14b, the least common multiple of 2,2b\times 7.

$$7b=b^{2}+25-49$$

Subtract 49 from 25 to get -24.

$$7b=b^{2}-24$$

Subtract b^{2} from both sides.

$$7b-b^{2}=-24$$

Add 24 to both sides.

$$7b-b^{2}+24=0$$

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.

$$-b^{2}+7b+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 7.

$$b=\frac{-7±\sqrt{49-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-7±\sqrt{49+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$b=\frac{-7±\sqrt{49+96}}{2\left(-1\right)}$$

Add 49 to 96.

$$b=\frac{-7±\sqrt{145}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-7±\sqrt{145}}{-2}$$

Now solve the equation b=\frac{-7±\sqrt{145}}{-2} when ± is plus. Add -7 to \sqrt{145}.

$$b=\frac{\sqrt{145}-7}{-2}$$

Divide -7+\sqrt{145} by -2.

$$b=\frac{7-\sqrt{145}}{2}$$

Now solve the equation b=\frac{-7±\sqrt{145}}{-2} when ± is minus. Subtract \sqrt{145} from -7.

$$b=\frac{-\sqrt{145}-7}{-2}$$

Divide -7-\sqrt{145} by -2.

$$b=\frac{\sqrt{145}+7}{2}$$

The equation is now solved.

$$b=\frac{7-\sqrt{145}}{2} b=\frac{\sqrt{145}+7}{2}$$

Steps for Completing the Square

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 14b, the least common multiple of 2,2b\times 7.

$$7b=b^{2}+25-49$$

Subtract 49 from 25 to get -24.

$$7b=b^{2}-24$$

Subtract b^{2} from both sides.

$$7b-b^{2}=-24$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-b^{2}+7b=-24$$

Divide both sides by -1.

$$\frac{-b^{2}+7b}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{7}{-1}b=-\frac{24}{-1}$$

Divide 7 by -1.

$$b^{2}-7b=-\frac{24}{-1}$$

Divide -24 by -1.

$$b^{2}-7b=24$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-7b+\left(-\frac{7}{2}\right)^{2}=24+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-7b+\frac{49}{4}=24+\frac{49}{4}$$

Add 24 to \frac{49}{4}.

$$b^{2}-7b+\frac{49}{4}=\frac{145}{4}$$

Factor b^{2}-7b+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{7}{2}\right)^{2}=\frac{145}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{7}{2}\right)^{2}}=\sqrt{\frac{145}{4}}$$

Simplify.

$$b-\frac{7}{2}=\frac{\sqrt{145}}{2} b-\frac{7}{2}=-\frac{\sqrt{145}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$b=\frac{\sqrt{145}+7}{2} b=\frac{7-\sqrt{145}}{2}$$