Solve -4.9x^2+307x+248=0

$$-4.9 { x }^{ 2 } +307x+248=0$$

Solve for x

$$x = \frac{3 \sqrt{275305} + 1535}{49} \approx 63.450723444$$

$$x=\frac{1535-3\sqrt{275305}}{49}\approx -0.79766222$$

Steps Using the Quadratic Formula

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.

$$-4.9x^{2}+307x+248=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 307 for b, and 248 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-307±\sqrt{307^{2}-4\left(-4.9\right)\times 248}}{2\left(-4.9\right)}$$

Square 307.

$$x=\frac{-307±\sqrt{94249-4\left(-4.9\right)\times 248}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$x=\frac{-307±\sqrt{94249+19.6\times 248}}{2\left(-4.9\right)}$$

Multiply 19.6 times 248.

$$x=\frac{-307±\sqrt{94249+4860.8}}{2\left(-4.9\right)}$$

Add 94249 to 4860.8.

$$x=\frac{-307±\sqrt{99109.8}}{2\left(-4.9\right)}$$

Take the square root of 99109.8.

$$x=\frac{-307±\frac{3\sqrt{275305}}{5}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8}$$

Now solve the equation x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8} when ± is plus. Add -307 to \frac{3\sqrt{275305}}{5}.

$$x=\frac{\frac{3\sqrt{275305}}{5}-307}{-9.8}$$

Divide -307+\frac{3\sqrt{275305}}{5} by -9.8 by multiplying -307+\frac{3\sqrt{275305}}{5} by the reciprocal of -9.8.

$$x=\frac{1535-3\sqrt{275305}}{49}$$

Now solve the equation x=\frac{-307±\frac{3\sqrt{275305}}{5}}{-9.8} when ± is minus. Subtract \frac{3\sqrt{275305}}{5} from -307.

$$x=\frac{-\frac{3\sqrt{275305}}{5}-307}{-9.8}$$

Divide -307-\frac{3\sqrt{275305}}{5} by -9.8 by multiplying -307-\frac{3\sqrt{275305}}{5} by the reciprocal of -9.8.

$$x=\frac{3\sqrt{275305}+1535}{49}$$

The equation is now solved.

$$x=\frac{1535-3\sqrt{275305}}{49} x=\frac{3\sqrt{275305}+1535}{49}$$

Steps for Completing the Square

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.

$$-4.9x^{2}+307x+248=0$$

Subtract 248 from both sides of the equation.

$$-4.9x^{2}+307x+248-248=-248$$

Subtracting 248 from itself leaves 0.

$$-4.9x^{2}+307x=-248$$

Divide both sides of the equation by -4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.9x^{2}+307x}{-4.9}=-\frac{248}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$x^{2}+\frac{307}{-4.9}x=-\frac{248}{-4.9}$$

Divide 307 by -4.9 by multiplying 307 by the reciprocal of -4.9.

$$x^{2}-\frac{3070}{49}x=-\frac{248}{-4.9}$$

Divide -248 by -4.9 by multiplying -248 by the reciprocal of -4.9.

$$x^{2}-\frac{3070}{49}x=\frac{2480}{49}$$

Divide -\frac{3070}{49}, the coefficient of the x term, by 2 to get -\frac{1535}{49}. Then add the square of -\frac{1535}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3070}{49}x+\left(-\frac{1535}{49}\right)^{2}=\frac{2480}{49}+\left(-\frac{1535}{49}\right)^{2}$$

Square -\frac{1535}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3070}{49}x+\frac{2356225}{2401}=\frac{2480}{49}+\frac{2356225}{2401}$$

Add \frac{2480}{49} to \frac{2356225}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3070}{49}x+\frac{2356225}{2401}=\frac{2477745}{2401}$$

Factor x^{2}-\frac{3070}{49}x+\frac{2356225}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1535}{49}\right)^{2}=\frac{2477745}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1535}{49}\right)^{2}}=\sqrt{\frac{2477745}{2401}}$$

Simplify.

$$x-\frac{1535}{49}=\frac{3\sqrt{275305}}{49} x-\frac{1535}{49}=-\frac{3\sqrt{275305}}{49}$$

Add \frac{1535}{49} to both sides of the equation.

$$x=\frac{3\sqrt{275305}+1535}{49} x=\frac{1535-3\sqrt{275305}}{49}$$