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$h(x)= 2x^{2} +11x +15$

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Integer inequalities with absolute values

Full solution

Q.

h(x)= 2x^{2} +11x +15

Factor or Use Quadratic Formula: To find the roots of the quadratic function $h(x) = 2x^2 + 11x + 15$ , we need to solve the equation $2x^2 + 11x + 15 = 0$ . We can attempt to factor the quadratic, or use the quadratic formula to find the roots.
Quadratic Formula Coefficients: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ . For our function $h(x)$ , $a = 2$ , $b = 11$ , and $c = 15$ .
Calculate Discriminant: First, we calculate the discriminant, which is $b^2 - 4ac$ . For our equation, the discriminant is $11^2 - 4(2)(15)$ .
Calculate Discriminant Result: Calculating the discriminant: $11^2 - 4(2)(15) = 121 - 120 = 1$ .
Use Quadratic Formula: Since the discriminant is positive, we will have two real and distinct roots. Now we can use the quadratic formula to find the roots.
Substitute Values: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula: $x = \frac{-11 \pm \sqrt{1}}{2 \times 2}$ .
Simplify Expression: Simplify the expression: $x = \frac{{-11 \pm 1}}{{4}}$ .
Find Two Roots: Now, we find the two roots by solving for $x$ with both the positive and negative values of the square root of the discriminant.
Positive Square Root: For the positive square root: $x = (-11 + 1) / 4 = -10 / 4 = -2.5$ .
Negative Square Root: For the negative square root: $x = ( -11 - 1 ) / 4 = -12 / 4 = -3$ .

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