Solve 2t^2+9t+7

$$2 t ^ { 2 } + 9 t + 7$$

Factor

$$\left(t+1\right)\left(2t+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2t^{2}+at+bt+7. To find a and b, set up a system to be solved.

$$a+b=9 ab=2\times 7=14$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 9.

$$a=2 b=7$$

Rewrite 2t^{2}+9t+7 as \left(2t^{2}+2t\right)+\left(7t+7\right).

$$\left(2t^{2}+2t\right)+\left(7t+7\right)$$

Factor out 2t in the first and 7 in the second group.

$$2t\left(t+1\right)+7\left(t+1\right)$$

Factor out common term t+1 by using distributive property.

$$\left(t+1\right)\left(2t+7\right)$$

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.

$$2t^{2}+9t+7=0$$

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.

$$t=\frac{-9±\sqrt{9^{2}-4\times 2\times 7}}{2\times 2}$$

Square 9.

$$t=\frac{-9±\sqrt{81-4\times 2\times 7}}{2\times 2}$$

Multiply -4 times 2.

$$t=\frac{-9±\sqrt{81-8\times 7}}{2\times 2}$$

Multiply -8 times 7.

$$t=\frac{-9±\sqrt{81-56}}{2\times 2}$$

Add 81 to -56.

$$t=\frac{-9±\sqrt{25}}{2\times 2}$$

Take the square root of 25.

$$t=\frac{-9±5}{2\times 2}$$

Multiply 2 times 2.

$$t=\frac{-9±5}{4}$$

Now solve the equation t=\frac{-9±5}{4} when ± is plus. Add -9 to 5.

$$t=-\frac{4}{4}$$

Divide -4 by 4.

$$t=-1$$

Now solve the equation t=\frac{-9±5}{4} when ± is minus. Subtract 5 from -9.

$$t=-\frac{14}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

$$t=-\frac{7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{7}{2} for x_{2}.

$$2t^{2}+9t+7=2\left(t-\left(-1\right)\right)\left(t-\left(-\frac{7}{2}\right)\right)$$

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

$$2t^{2}+9t+7=2\left(t+1\right)\left(t+\frac{7}{2}\right)$$

Add \frac{7}{2} to t by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2t^{2}+9t+7=2\left(t+1\right)\times \frac{2t+7}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2t^{2}+9t+7=\left(t+1\right)\left(2t+7\right)$$

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.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 +\frac{9}{2}x +\frac{7}{2} = 0$$

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

$$r + s = -\frac{9}{2} rs = \frac{7}{2}$$

Two numbers r and s sum up to -\frac{9}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{2} = -\frac{9}{4}. 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>

$$r = -\frac{9}{4} - u s = -\frac{9}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{2}

$$(-\frac{9}{4} - u) (-\frac{9}{4} + u) = \frac{7}{2}$$

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

$$\frac{81}{16} - u^2 = \frac{7}{2}$$

Simplify the expression by subtracting \frac{81}{16} on both sides

$$-u^2 = \frac{7}{2}-\frac{81}{16} = -\frac{25}{16}$$

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

$$u^2 = \frac{25}{16} u = \pm\sqrt{\frac{25}{16}} = \pm \frac{5}{4}$$

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

$$r =-\frac{9}{4} - \frac{5}{4} = -3.500 s = -\frac{9}{4} + \frac{5}{4} = -1$$

Evaluate

$$\left(t+1\right)\left(2t+7\right)$$