Question

Solve the following quadratic equations by factorization: 5² + 9m – 2 = 0​

Answer 1

EXPLANATION.

Quadratic equation,

⇒ F(x) = 5m² + 9m - 2.

As we know that,

Factorizes the equation into middle term split, we get.

⇒ 5m² + 9m - 2 = 0.

⇒ 5m² + 10m - m - 2 = 0.

⇒ 5m(m + 2) - 1(m + 2) = 0.

⇒ (5m - 1)(m + 2) = 0.

⇒ m = 1/5 & m = -2.

                                                                                       

MORE INFORMATION.

Nature of roots.

The term b² - 4ac is called discriminant of the equation. it is denoted by Δ or D.

(A) = Suppose a, b, c ∈ R and a ≠ 0 then,

(1) = If D > 0 Roots are real and unequal.

(2) = If D = 0 Roots are real and equal and each equal to -b/2a.

(3) = If D < 0 Roots are imaginary and unequal or complex conjugate.

(B) = Suppose a, b, c ∈ Q, a ≠ 0 then,

(1) = If D > 0 & D is perfect square ⇒ Roots are unequal & rational.

(2) = If D > 0 & D is not perfect square ⇒ Roots are irrational & unequal.

Answer 2

Step-by-step explanation:

${\large{\bold{\rm{\underline{Let's\ understand\ the\ concept\ 1^{st}:-}}}}}$

★ Here the concept of Quadratic Equation has been used. We see that we are given a quadratic equation and we have to solve it. The best and easiest method to solve this is the method of Quadratic Formula. In this method we shall split the the middle term into two terms which will provide us the common term and thus finally given us the two value. These will be the given answer.

${\large{\bold{\rm{\underline{Given\ that:-}}}}}$

~ Quadratic equation:-

$\sf {\bigstar F(x)\ =\ 5m^2\ +\ 9m\ -\ 2.}$

${\large{\bold{\rm{\underline{Solution:-}}}}}$

★ m = 1/5 and m = -2.

${\large{\bold{\rm{\underline{Full\ solution:-}}}}}$

~ By splitting the equation into middle term, we get:-

$\sf : \implies {5m^2\ +\ 9m\ -\ 2\ =\ 0}$

$\sf : \implies {5m^2\ +\ 10m\ -\ m\ -\ 2\ =\ 0}$

$\sf : \implies {5m(m\ +\ 2)\ -\ 1(m\ +\ 2)\ =\ 0}$

$\sf : \implies {(5m\ -\ 1)(m\ +\ 2)\ =\ 0}$

$\bf : \implies {\blue{m\ =\ 1/5\ and\ m\ =\ -2.}}$

${\large{\bold{\bf{\underline{\underline{More\ to\ know:-}}}}}}$

~ Following ways help us to solve a quadratic equation:-

» First we need to express the given equation in the general form of the quadratic equation ax² + bx + c = 0, then,

» We need to factorize the left side of the quadratic equation,

» Now express each of the two factor equals to 0 and solve them,

» The two solutions are called the roots of the given quadratic equation.

▪︎Note:-

• If b ≠ 0 and c = 0, one root of the quadratic equation is always zero.

~ Properties of quadratic equation:-

» It has two solutions.

» The highest degree of the equation is 2.

» The graph of the equation intersects x - axis twice.

» Has only one variable to find solution.