Solve 0x^2+14x+33=0

$$0 x ^ { 2 } + 14 x + 33 = 0$$

Solve for x

$$x = -\frac{33}{14} = -2\frac{5}{14} \approx -2.357142857$$

Steps for Solving Linear Equation

Anything times zero gives zero.

$$0+14x+33=0$$

Add 0 and 33 to get 33.

$$33+14x=0$$

Subtract 33 from both sides. Anything subtracted from zero gives its negation.

$$14x=-33$$

Divide both sides by 14.

$$x=\frac{-33}{14}$$

Fraction \frac{-33}{14} can be rewritten as -\frac{33}{14} by extracting the negative sign.

$$x=-\frac{33}{14}$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +14x +33 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -14 rs = 33$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 33

$$(-7 - u) (-7 + u) = 33$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$49 - u^2 = 33$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 33-49 = -16$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-7 - 4 = -11 s = -7 + 4 = -3$$