Respuesta :

the quadratic equation:
4x^2 + 34x + 60 = 0
4x^2 + 24x + 10x + 60 = 0
4x(x + 6) + 10(x + 6)
(4x + 10) (x + 6)
x = - 6 , - 5/2
the answer is : c. -6, -5/2

hope this help

The correct option is [tex]\boxed{\bf option\ C }[/tex] i.e., [tex]\boxed{\bf \left(-6,\dfrac{-5}{2}\right)}[/tex]

Further explanation:  

The standard form of the quadratic equation is as follows:

[tex]\boxed{ax^{2}+bx+c=0}[/tex]

In the above equation [tex]a,b\text{ and }c[/tex]  are real numbers.

The solution of the quadratic equation can be evaluated by the quadratic rule with the use of discriminant formula and by middle term splitting formula.

The quadratic formula to obtain the roots of a quadratic equation [tex]ax^{2}+bx+c=0[/tex] is as follows:

[tex]\boxed{x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}[/tex]

In the above equation the term [tex]b^{2}-4ac[/tex] is called the discriminant and the expression for the discriminant is as follows:

[tex]\boxed{D=b^{2}-4ac}[/tex]

The discriminant is a parameter which is used to determine the nature of the roots.

Given:

The quadratic equation is given as follows:

[tex]\boxed{4x^{2}+34x+60=0}[/tex]

Calculation:

The given equation is [tex]4x^{2}+34x+60=0[/tex].

On comparing the given equation with standard quadratic equation, it is observed that the value of [tex]a,b\text{ and }c[/tex] are as follows:

[tex]\boxed{\begin{aligned}a&=4\\b&=34\\c&=60\end{aligned}}[/tex]  

We are solving this equation by the quadratic formula.

Step 1:

First find the discriminant of the equation to check the nature of the roots.

[tex]\begin{aligned}D&=(34)^{2}-(4\cdot 4\cdot 60)\\&=1156-960\\&=196\end{aligned}[/tex]  

Here, the value of discriminant is positive. Therefore, the roots are real.

Step 2:

Now, use the quadratic formula to obtain the value of real roots.

[tex]\begin{aligned}x&=\dfrac{-34\pm \sqrt{196}}{2\cdot 4}\\x&=\dfrac{-34\pm 14}{8}\\x&=\dfrac{-34+14}{8}\ \ \text{or} \ \ x=\dfrac{-34-14}{8}\\x&=-\dfrac{5}{2}\ \ \ \ \ \ \ \ \ \ \text{or}\ \ x=-6\end{aligned}[/tex]  

From the above calculation it is concluded that the roots of the given quadratic equation are [tex]\frac{-5}{2}[/tex] and [tex]-6[/tex].

Therefore, the correct option is [tex]\boxed{\bf\ option C}[/tex] i.e., [tex]\boxed{\left(-6,\dfrac{-5}{2}\right)}[/tex]

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Quadratic equation

Keywords: Quadratic equation, polynomials, discriminant, roots, solutions, quadratic rule, real roots, imaginary roots, middle term splitting formula, factors, positive , negative , greater than , less than, standard equation.

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