Discriminant Formula in Quadratic Equations

Algebra can be defined as the branch of mathematics which deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for the purpose of studying situations involving variables. It also has various sub-branches such as linear algebra, advanced algebra, commutative algebra, etc.

What are Quadratic Equations?

The degree of a polynomial is the highest power of the variable in it. A quadratic equation can be defined as a polynomial equation that has a degree of 2. 

ax² + bx + c = 0

Where a and b are the coefficients, x is the unknown variable and c is the constant, and a ≠ 0.

What is Discriminant?

The discriminant is a value derived from the coefficients of a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots (solutions) of the quadratic equation. The discriminant is denoted by the symbol D.

Discriminant Formula for Solving a Quadratic Equation

Since a quadratic equation has a degree of 2, therefore it will have two solutions. Therefore there would be two values of the variable x for which the equation is satisfied. According to the discriminant formula, a quadratic equation of the form ax² + bx + c = 0 has two roots, given by:

x = \frac{-b±\sqrt{D}}{2a} , 

Where D = b² − 4ac.

The ± signs indicate two distinct solutions to the equation. If the discriminant comes out to be negative, then the given equation does not have any real roots.

Derivation of Discriminant Formula

It can be derived by using completing the square method and then solving the equation for x.

ax² + bx + c = 0

Divide both sides by a.

⇒ x² + \frac{b}{a}x + \frac{c}{a}   = 0

⇒ x² + \frac{b}{a}x   = -\frac{c}{a}

Add (\frac{b}{2a})^2   to both sides.

⇒ x² + \frac{b}{a}x + (\frac{b}{2a})^2 = (\frac{b}{2a})^2 -\frac{c}{a}

Apply the identity: a² + b² + 2ab = (a + b)²

⇒ (x+\frac{b}{2a})^2   = (\frac{b}{2a})^2 -\frac{c}{a}

Take square root on both sides.

⇒ x + \frac{b}{2a}   = ±\sqrt{(\frac{b}{2a})^2 -\frac{c}{a}}

⇒ x = -\frac{b}{2a} ±\sqrt{(\frac{b}{2a})^2 -\frac{c}{a}}

⇒ x = \frac{-b±\sqrt{b^2-4ac}}{2a}

Sample Questions on Discriminant Formula

Question 1. Solve for x: x² = −2x + 2 using discriminant formula.

Solution:

Given: x² = −2x + 2 or,  x² + 2x − 2 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 1,  b = 2, c = −2.

⇒  x = \frac{-2±\sqrt{(2^2-4(1)(-2))}}{2(1)}

⇒  x = \frac{-2+\sqrt{4+8}}{2}

⇒  x = (−1 + √3),(−1 – √3).

Question 2. Solve for y: 2y² − 8y − 10 = 0 using discriminant formula.

Solution:

Given: 2y² − 8y − 10 = 0

According to discriminant formula, y = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 2,  b = −8, c = −10.

⇒  y = \frac{8±\sqrt{((-8)^2-4(2)(-10))}}{2(2)}

⇒  y = \frac{8±\sqrt{144}}{2(2)}

⇒  y = 4, −1.

Question 3. Solve for x: 2x² − 7x + 3 = 0 using discriminant formula.

Solution:

Given: 2x² − 7x + 3 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 2,  b = −7, c = 3.

⇒  x = \frac{7±\sqrt{(-7)^2-4(2)(3)}}{2(2)}

⇒  x = \frac{7±5}{4}

⇒  x = 3, 1/2.

Question 4. Solve for x: x² − 2x + 3 = 0 using discriminant formula.

Solution:

Given: x² − 2x + 3 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 1,  b = −2, c = 3.

⇒  x = \frac{2±\sqrt{(-2)^2-4(1)(3)}}{2(1)}

⇒  x = \frac{2±\sqrt{-8}}{2}

Since the value of the discriminant is less than zero (D = −8 < 0), the given quadratic equation has no real solution.

Question 5. Solve for x: x² + 5x + 4 = 0 using discriminant formula.

Solution:

Given: x² + 5x + 4 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 1,  b = 5, c = 4.

⇒  x = \frac{-5±\sqrt{(5)^2-4(1)(4)}}{2(1)}

⇒  x = \frac{-5±3}{2}

⇒  x = -1, -4.

Question 6. Solve for x: 6x² − x − 15 = 0 using discriminant formula.

Solution:

Given: 6x² − x − 15 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 6,  b = −1, c = −15.

⇒  x = \frac{1±\sqrt{(-1)^2-4(6)(-15)}}{2(6)}

⇒  x = \frac{1±19}{12}

⇒  x = 5/3, −3/2.

Question 7. Solve for x: x² + 4x + 9 = 0 using discriminant formula.

Solution:

Given: x² + 4x + 9 = 0

According to discriminant formula, x = \frac{-b±\sqrt{(b^2-4ac)}}{2a}

Here, a = 1,  b = 4, c = 9.

⇒  x = \frac{-4±\sqrt{(4)^2-4(1)(9)}}{2(1)}

⇒  x = \frac{-4±\sqrt{-20}}{2}

Since the value of the discriminant is less than zero (D = −20 < 0), the given quadratic equation has no real solution.