Solve 14x+10,5=x^2+1,5x

$$14 x + 10,5 = x ^ { 2 } + 1,5 x$$

Solve for x

$$x = \frac{\sqrt{793} + 25}{4} \approx 13.29006392$$

$$x=\frac{25-\sqrt{793}}{4}\approx -0.79006392$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$14x+10,5-x^{2}=1,5x$$

Subtract 1,5x from both sides.

$$14x+10,5-x^{2}-1,5x=0$$

Combine 14x and -1,5x to get 12,5x.

$$12,5x+10,5-x^{2}=0$$

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.

$$-x^{2}+12,5x+10,5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12,5 for b, and 10,5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12,5±\sqrt{12,5^{2}-4\left(-1\right)\times 10,5}}{2\left(-1\right)}$$

Square 12,5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-12,5±\sqrt{156,25-4\left(-1\right)\times 10,5}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12,5±\sqrt{156,25+4\times 10,5}}{2\left(-1\right)}$$

Multiply 4 times 10,5.

$$x=\frac{-12,5±\sqrt{156,25+42}}{2\left(-1\right)}$$

Add 156,25 to 42.

$$x=\frac{-12,5±\sqrt{198,25}}{2\left(-1\right)}$$

Take the square root of 198,25.

$$x=\frac{-12,5±\frac{\sqrt{793}}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2}$$

Now solve the equation x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2} when ± is plus. Add -12,5 to \frac{\sqrt{793}}{2}.

$$x=\frac{\sqrt{793}-25}{-2\times 2}$$

Divide \frac{-25+\sqrt{793}}{2} by -2.

$$x=\frac{25-\sqrt{793}}{4}$$

Now solve the equation x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2} when ± is minus. Subtract \frac{\sqrt{793}}{2} from -12,5.

$$x=\frac{-\sqrt{793}-25}{-2\times 2}$$

Divide \frac{-25-\sqrt{793}}{2} by -2.

$$x=\frac{\sqrt{793}+25}{4}$$

The equation is now solved.

$$x=\frac{25-\sqrt{793}}{4} x=\frac{\sqrt{793}+25}{4}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$14x+10,5-x^{2}=1,5x$$

Subtract 1,5x from both sides.

$$14x+10,5-x^{2}-1,5x=0$$

Combine 14x and -1,5x to get 12,5x.

$$12,5x+10,5-x^{2}=0$$

Subtract 10,5 from both sides. Anything subtracted from zero gives its negation.

$$12,5x-x^{2}=-10,5$$

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.

$$-x^{2}+12,5x=-10,5$$

Divide both sides by -1.

$$\frac{-x^{2}+12,5x}{-1}=-\frac{10,5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12,5}{-1}x=-\frac{10,5}{-1}$$

Divide 12,5 by -1.

$$x^{2}-12,5x=-\frac{10,5}{-1}$$

Divide -10,5 by -1.

$$x^{2}-12,5x=10,5$$

Divide -12,5, the coefficient of the x term, by 2 to get -6,25. Then add the square of -6,25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12,5x+\left(-6,25\right)^{2}=10,5+\left(-6,25\right)^{2}$$

Square -6,25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-12,5x+39,0625=10,5+39,0625$$

Add 10,5 to 39,0625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-12,5x+39,0625=49,5625$$

Factor x^{2}-12,5x+39,0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6,25\right)^{2}=49,5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6,25\right)^{2}}=\sqrt{49,5625}$$

Simplify.

$$x-6,25=\frac{\sqrt{793}}{4} x-6,25=-\frac{\sqrt{793}}{4}$$

Add 6,25 to both sides of the equation.

$$x=\frac{\sqrt{793}+25}{4} x=\frac{25-\sqrt{793}}{4}$$