Solve G(x)=-6x^2+170x+300

$$G ( x ) = - 6 x ^ { 2 } + 170 x + 300$$

Factor

$$2\left(30-x\right)\left(3x+5\right)$$

Solution Steps

Factor out 2.

$$2\left(-3x^{2}+85x+150\right)$$

Consider -3x^{2}+85x+150. Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx+150. To find a and b, set up a system to be solved.

$$a+b=85 ab=-3\times 150=-450$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -450.

$$-1,450 -2,225 -3,150 -5,90 -6,75 -9,50 -10,45 -15,30 -18,25$$

Calculate the sum for each pair.

$$-1+450=449 -2+225=223 -3+150=147 -5+90=85 -6+75=69 -9+50=41 -10+45=35 -15+30=15 -18+25=7$$

The solution is the pair that gives sum 85.

$$a=90 b=-5$$

Rewrite -3x^{2}+85x+150 as \left(-3x^{2}+90x\right)+\left(-5x+150\right).

$$\left(-3x^{2}+90x\right)+\left(-5x+150\right)$$

Factor out 3x in the first and 5 in the second group.

$$3x\left(-x+30\right)+5\left(-x+30\right)$$

Factor out common term -x+30 by using distributive property.

$$\left(-x+30\right)\left(3x+5\right)$$

Rewrite the complete factored expression.

$$2\left(-x+30\right)\left(3x+5\right)$$

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.

$$-6x^{2}+170x+300=0$$

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.

$$x=\frac{-170±\sqrt{170^{2}-4\left(-6\right)\times 300}}{2\left(-6\right)}$$

Square 170.

$$x=\frac{-170±\sqrt{28900-4\left(-6\right)\times 300}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-170±\sqrt{28900+24\times 300}}{2\left(-6\right)}$$

Multiply 24 times 300.

$$x=\frac{-170±\sqrt{28900+7200}}{2\left(-6\right)}$$

Add 28900 to 7200.

$$x=\frac{-170±\sqrt{36100}}{2\left(-6\right)}$$

Take the square root of 36100.

$$x=\frac{-170±190}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-170±190}{-12}$$

Now solve the equation x=\frac{-170±190}{-12} when ± is plus. Add -170 to 190.

$$x=\frac{20}{-12}$$

Reduce the fraction \frac{20}{-12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{3}$$

Now solve the equation x=\frac{-170±190}{-12} when ± is minus. Subtract 190 from -170.

$$x=-\frac{360}{-12}$$

Divide -360 by -12.

$$x=30$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{3} for x_{1} and 30 for x_{2}.

$$-6x^{2}+170x+300=-6\left(x-\left(-\frac{5}{3}\right)\right)\left(x-30\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-6x^{2}+170x+300=-6\left(x+\frac{5}{3}\right)\left(x-30\right)$$

Add \frac{5}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-6x^{2}+170x+300=-6\times \frac{-3x-5}{-3}\left(x-30\right)$$

Cancel out 3, the greatest common factor in -6 and 3.

$$-6x^{2}+170x+300=2\left(-3x-5\right)\left(x-30\right)$$

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.

$$x ^ 2 -\frac{85}{3}x -50 = 0$$

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

$$r + s = \frac{85}{3} rs = -50$$

Two numbers r and s sum up to \frac{85}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{85}{3} = \frac{85}{6}. 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>

$$r = \frac{85}{6} - u s = \frac{85}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -50

$$(\frac{85}{6} - u) (\frac{85}{6} + u) = -50$$

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

$$\frac{7225}{36} - u^2 = -50$$

Simplify the expression by subtracting \frac{7225}{36} on both sides

$$-u^2 = -50-\frac{7225}{36} = -\frac{9025}{36}$$

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

$$u^2 = \frac{9025}{36} u = \pm\sqrt{\frac{9025}{36}} = \pm \frac{95}{6}$$

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

$$r =\frac{85}{6} - \frac{95}{6} = -1.667 s = \frac{85}{6} + \frac{95}{6} = 30$$

Evaluate

$$300+170x-6x^{2}$$