Select the two values of x that are roots of this equation. 2x2 + 7x + 6 = 0

Mathematics

2 answers:

julia-pushkina [17]2 years ago

8 0

Answer:

$\large {\textsf{A and C}}\ \implies x_1=-2,\ x_2=-\dfrac{3}{2}}$

Step-by-step explanation:

In this problem, we can use two methods to solve for the values of x (roots) of the given equation. Those methods are: using the quadratic formula and factoring.

Standard Form of a Quadratic Equation: ax² + bx + c = 0, where a ≠ 0

Given equation: 2x² + 7x + 6 = 0

⇒ a = 2, b = 7, c = 6

Method 1: Using the Quadratic Formula

Quadratic Formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step 1: Substitute the values of a, b, and c into the quadratic formula.

$\implies x=\dfrac{-7\pm\sqrt{7^2-4(2)(6)}}{2(2)}\\\\$

Step 2: Simplify

$\implies x=\dfrac{-7\pm\sqrt{\bold{7^2}-4(2)(6)}}{\bold{2(2)}}\\\\\implies x=\dfrac{-7\pm\sqrt{49-4\bold{(2)(6)}}}{4}\\\\\implies x=\dfrac{-7\pm\sqrt{49\bold{-4(12)}}}{4}\\\\\implies x=\dfrac{-7\pm\sqrt{\bold{49-48}}}{4}\\\\\implies x=\dfrac{-7\pm\sqrt{\bold{1}}}{4}\\\\\implies x=\dfrac{-7\pm1}{4}$

Step 3: Separate into two possible cases and solve for the values of x.

$\implies x_1=\dfrac{-7-1}{4}\implies \dfrac{-8}{4}\implies \boxed{-2}\\\\\implies x_2=\dfrac{-7+1}{4}\implies \dfrac{-6}{4}\implies\boxed{-\dfrac{3}{2}}$

Method 2: Solve by Factoring

In order to be able to solve this equation by factoring, let's rewrite the middle term by finding the factors that give a product of the first and last terms (a • c = 12) and give us the sum of the middle term (b = 7).

Factors that give a product of a • c: 4 • 3 = 12

Factors that give a sum of b: 4 + 3 = 7

Step 1: Rewrite the given equation with those factors.

2x² + 7x + 6 = 0

⇒ 2x² + 4x + 3x + 6 = 0

Step 2: Factor out 2x and 3.

(2x² + 4x) + (3x + 6) = 0

⇒ 2x(x + 2) + 3(x + 2) = 0 [ Factor out the the common factor. ]

⇒ (2x + 3)(x + 2) = 0

Step 3: Apply the Zero-Product Property (if m•n = 0, then m = 0 or n = 0).

a) 2x + 3 = 0 ⇒ 2x = -3 ⇒ x = -³⁄₂

b) x + 2 = 0 ⇒ x = -2