CBSE Class 9 Mathematics Quadratic Equations Notes Set 06

Quadratic Equations and Their Roots

An equation of the form \( ax^2+bx+c=0 \) with \( a\ne0 \) is a quadratic equation. Completing the square gives the roots \( x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \). Many quadratics can be solved faster by splitting the middle term: find two numbers whose product is \( ac \) and whose sum is \( b \), then factorise. For \( 6x^2-x-2=0 \) the numbers are \( -4 \) and \( 3 \), giving \( (2x+1)(3x-2)=0 \) and roots \( -\dfrac12 \) and \( \dfrac23 \).

Nature of the Roots

The expression \( \Delta=b^2-4ac \) is the discriminant. If \( \Delta\lt0 \) the roots are not real. If \( \Delta=0 \) the roots are real and equal, each \( -\dfrac b{2a} \). If \( \Delta>0 \) the roots are real and unequal, and they are rational when \( \Delta \) is a perfect square (for rational coefficients). Example: \( x^2+5x+5=0 \) has \( \Delta=5 \), so its roots are real, unequal and irrational; \( 4x^2+12x+9=0 \) has \( \Delta=0 \) and the repeated root \( -\dfrac32 \).

Sum and Product of Roots

If \( \alpha \) and \( \beta \) are the roots of \( ax^2+bx+c=0 \) then \( \alpha+\beta=-\dfrac ba \) and \( \alpha\beta=\dfrac ca \). Conversely, the equation with roots \( \alpha,\beta \) is \( x^2-(\alpha+\beta)x+\alpha\beta=0 \). Expressions such as \( \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta \) and \( \alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta) \) can be found without solving the equation.

Graphs of Quadratic Functions

The graph of \( y=ax^2+bx+c \) is a parabola. It opens upwards if \( a>0 \) (with a least value at the turning point) and downwards if \( a\lt0 \) (with a greatest value). The real roots of the equation are the x-coordinates of the points where the curve meets the x-axis. Completing the square shows the turning point: \( y=x^2+2x-8=(x+1)^2-9 \) has least value \( -9 \) at \( x=-1 \), and its roots are \( -4 \) and \( 2 \). Another way to solve \( x^2-x-6=0 \) is to draw \( y=x^2 \) and \( y=x+6 \); they meet at \( x=-2 \) and \( x=3 \).

Word Problems

Form the equation from the story, simplify to the standard form and reject any root that makes no sense (a negative length, number of days or speed). If the speed of one car is \( x \) km/h and a second goes 2 km/h faster, the difference in times for distances 81 km and 85 km being a quarter of an hour gives \( \dfrac{81}x-\dfrac{85}{x+2}=\dfrac14 \), that is \( (x-18)(x+36)=0 \), so the speeds are 18 and 20 km/h.

Worked Examples

Example 1. Solve \( x+5-\dfrac8{x+5}=7 \).
Put \( u=x+5 \): \( u^2-7u-8=0 \), so \( u=8 \) or \( u=-1 \), giving \( x=3 \) or \( x=-6 \).

Example 2. Form the equation whose roots are 3 and \( -2 \).
Sum \( =1 \), product \( =-6 \), so \( x^2-x-6=0 \).

Example 3. Find \( k \) so that \( 2x^2-3x+k=0 \) has equal roots.
\( \Delta=9-8k=0 \), so \( k=\dfrac98 \).