Solve -3x^2-15x-21

$$- 3 x ^ { 2 } - 15 x - 21$$

Factor

$$3\left(-x^{2}-5x-7\right)$$

Solution Steps

Factor out 3. Polynomial -x^{2}-5x-7 is not factored since it does not have any rational roots.

$$3\left(-x^{2}-5x-7\right)$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}-15x-21=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-3\right)\left(-21\right)}}{2\left(-3\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-3\right)\left(-21\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-15\right)±\sqrt{225+12\left(-21\right)}}{2\left(-3\right)}$$

Multiply 12 times -21.

$$x=\frac{-\left(-15\right)±\sqrt{225-252}}{2\left(-3\right)}$$

Add 225 to -252.

$$x=\frac{-\left(-15\right)±\sqrt{-27}}{2\left(-3\right)}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$-3x^{2}-15x-21$$

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.

$$x ^ 2 +5x +7 = 0$$

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

$$r + s = -5 rs = 7$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{2}. 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>

$$r = -\frac{5}{2} - u s = -\frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 7

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = 7$$

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

$$\frac{25}{4} - u^2 = 7$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = 7-\frac{25}{4} = \frac{3}{4}$$

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

$$u^2 = -\frac{3}{4} u = \pm\sqrt{-\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}i$$

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

$$r =-\frac{5}{2} - \frac{\sqrt{3}}{2}i = -2.500 - 0.866i s = -\frac{5}{2} + \frac{\sqrt{3}}{2}i = -2.500 + 0.866i$$

Evaluate

$$-3x^{2}-15x-21$$