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Vsevolod [243]
2 years ago
11

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

<u>Given equation:</u> 2x² + 7x + 6 = 0

⇒ a = 2, b = 7, c = 6

Method 1: Using the Quadratic Formula

<u>Quadratic Formula:</u> x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

<u>Step 1:</u> Substitute the values of <em>a</em>, <em>b</em>, and <em>c</em> into the quadratic formula.

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

<u>Step 2:</u> 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}

<u>Step 3:</u> 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 <em>middle term</em> 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: <em>4 • 3</em> = 12

Factors that give a sum of b: <em>4 + 3</em> = 7

<u>Step 1:</u> Rewrite the given equation with those factors.

2x² + 7x + 6 = 0

⇒ 2x² + 4x + 3x + 6 = 0

<u>Step 2:</u> 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

<u>Step 3:</u> Apply the <em>Zero-Product Property</em> (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

Therefore, the two roots of this equation are x = -³⁄₂ and x = -2.

Learn more about quadratic equations here:

brainly.com/question/27989834

brainly.com/question/27638369

Anna007 [38]2 years ago
7 0

Answer:

x = -3/2, -2

Step-by-step explanation:

2x^2 + 7x + 6 = 0

(2x + 3)(x + 2) = 0

(2x + 3)

2x = -3

x = -3/2

(x + 2)

x = -2

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Answer:

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Step-by-step explanation:

Convert 1 8/11 into an improper fraction:  1 8/11 = 19/11.

Next, multiply this 19/11 by 3, obtaining 57/11.

This mixed numer is an improper fraction.  

If you wish to go further, convert 57/11 into a mixed number:

57/11 = 5 2/11

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Answer:

a+b is divisible by c since d and e are also whole numbers.

b-a is divisible by c since d and e are also whole numbers.

Step-by-step explanation:

Let a, b whole number that are divisible by c. Then, we have that d = \frac{a}{c} and e = \frac{b}{c}, which are also whole numbers. We proceed to prove that a + b is divisible by c:

1) d = \frac{a}{c}, e = \frac{b}{c}, a, b , c, d, e \in \mathbb{Z} Given

2) d + e = \frac{a}{c}+\frac{b}{c} Compatibility with addition

3) d+e = a\cdot c^{-1}+b\cdot c^{-1} Definition of division

4) d + e = c^{-1}\cdot (a+b) Commutative and distributive properties

5) d+e = (a+b) \cdot c^{-1} Commutative properties

6) d+e = \frac{a+b}{c} Definition of division/Result

Therefore, a+b is divisible by c since d and e are also whole numbers.

We proceed to prove that b-a is divisible by c:

1) d = \frac{a}{c}, e = \frac{b}{c}, a, b , c, d, e \in \mathbb{Z} Given

2) d\cdot (-1) = \frac{a}{c}\cdot (-1) Compatibility with multiplication

3) e + d\cdot (-1) = \frac{b}{c}+\frac{a}{c}\cdot (-1) Compatibility with addition

4) e + d\cdot (-1) = b\cdot c^{-1}+(a\cdot c^{-1})\cdot (-1) Definition of division

5) e + d\cdot (-1) = b\cdot c^{-1} + [a\cdot (-1)]\cdot c^{-1} Associative and commutative properties

6) e + d\cdot (-1) = c^{-1}\cdot [b+a\cdot (-1)] Commutative and distributive properties.

7) e+d\cdot (-1) = [b+a\cdot (-1)]\cdot c^{-1} Commutative properties.

8) e + d\cdot (-1) = \frac{b+a\cdot (-1)}{c} Definition of division

9) e-d = \frac{b-a}{c} (-1)\cdot a = -a/Definition of subtraction

Therefore, b-a is divisible by c since d and e are also whole numbers.

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