Class 10 Mathematics

An equation of the form ax²  + bx + c = 0 where a not equal to 0 is known as quadratic equation. a, b and c are constants. The name Quadratic comes from “quad” meaning square, because the variable gets squared (like x²).
It is also called an “Equation of Degree 2” (because of the “2” on the x)

The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.

1. Standard form of quadratic equation:  ax^₂ + bx + c = 0 where a, b and c are real numbers, x is a variable and a ≠ 0
2. Pure form of a quadratic equation: ax² + c = 0, where a and c are real numbers and a ≠ 0, c ≠ 0
3. A quadratic equation can have a maximum of 2 roots (zeroes)
4. Use the quadratic formula span id="MathJax-Element-1Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b(b24ac)2a">−b±(b2–4ac)√2a−b±(b2–4ac)2a to find the value(s) of x.
5. For the quadratic equation ax² + bx + c = 0, b² – 4ac is known as the discriminant (D)

Methods of Solving a Quadratic Equation

• Completing the Square
• Factorization
  • Factorize into product of 2 linear factors
  • Equate each factor to 0 and solve for x
• Quadratic Formula
  • span id="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="x=b(b24ac)2a">x=−b±(b2–4ac)√2ax=−b±(b2–4ac)2a, if span d="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b24ac0">b2−4ac≥0b2−4ac≥0, to find for x.
  • span id="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b24ac">b2−4acb2−4ac is known as the discriminant (D or Δ).

2. Solution of a Quadratic Equation by Factorisation

There are many ways to solve quadratic equations, such as taking square root, completing the square, and using the Quadratic Formula. Another method of solving quadratic equations is by factorisation, i.e., by writing the quadratic equation as a product of two linear factors and then equating each of the linear factors to zero.

Steps to solve quadratic equations by factorisation

1. Factorize the quadratic equation and write it as product of 2 linear factors. This is done mostly by splitting the term containing x.
2. Equate each factor to 0 and solve for x.

3. Solution of a Quadratic Equation by Completing the Square

At times, you might encounter quadratic equations that do not factor easily and are not in a form where square root can be taken. In such cases, the method of completing the square can be used.

Steps to solve a quadratic equation by completing the square

Complete the square by following the steps shown in the flowchart and find the value(s) of x.

Note:  with reference to the flow chart , if a≠1, we can also multiply  the entire equation by a. Then we only, shift constant term to RHS, and follow the  remaining steps shown in the block diagram.

 

4. Nature of Roots

The quadratic formula is an important formula that can be used to find the roots of any quadratic equation of the form ax² + bx + c = 0. The advantage of using the quadratic formula to solve the equation compared to other methods is that the quadratic formula always works, no matter how complex the equation.

The quadratic formula requires finding the square root of an expression, called the discriminant. The discriminant is useful to determine the nature of the roots of the equation before even attempting to solve it.

Discriminant and the Quadratic Formula

In a quadratic equation, the discriminant helps you find the number of real solutions to a quadratic equation.

For a quadratic equation ax² + bx + c = 0, the nature of roots is dependent on the value of the discriminant D, which is calculated as D = b² – 4ac

• If D > 0 and D is a perfect square, the roots are distinct, real and rational
  if D > 0 and D is not a perfect square, the roots are distinct, real but irrational
• If D 0, here are no real roots
• If D = 0, there are two equal, real roots

x = spanid="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="bD2a">−b±√D2a−b±√D2a is called the quadratic formula.

Relationship between roots and coefficients

When the two roots α and β are given, the quadratic equation ax² + bx + c = 0 can be derived using the formula:
x² – (sum of the roots)x  + (product of roots) = 0

where sum of the roots = span id="MathJax-Element--Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="+=ba">α+β=–baα+β=–ba

and product of the roots = span id="MathJax-Element-7-rame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="=ca">αβ=caαβ=ca