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4.03 The quadratic formula

Lesson

The quadratic formula gives the solutions to any quadratic equation in one variable. It can be used to solve quadratic equations which cannot be solved by factorisation.

Before we can use the quadratic formula, we have to rearrange the quadratic equation into the form

$Ax^2+Bx+C=0$Ax2+Bx+C=0

where $A$A, $B$B, and $C$C are any number and $A\ne0$A0.

Once the equation is in this form, the solutions are given by the quadratic formula:

$x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}$x=B±B24AC2A

Worked example

Solve $x^2+9x-7=0$x2+9x7=0 for $x$x.

Think: There are no integers which add to give $9$9 and multiply to give $-7$7 so we cannot factorise the left hand side.

However, we already have the equation in the right form to use the quadratic formula. Here, $A=1$A=1, $B=9$B=9 and $C=-7$C=7.

Do: We can substitute these values into the quadratic formula.

$x$x $=$= $\frac{-B\pm\sqrt{B^2-4AC}}{2A}$B±B24AC2A

The quadratic formula

  $=$= $\frac{-9\pm\sqrt{9^2-4\times1\times\left(-7\right)}}{2\times1}$9±924×1×(7)2×1

Substituting $A=1$A=1, $B=9$B=9, and $C=-7$C=7

  $=$= $\frac{-9\pm\sqrt{109}}{2}$9±1092

Simplifying the expression

So the solutions to the equation are $x=\frac{-9\pm\sqrt{109}}{2}$x=9±1092

Reflect: Notice that we ended up with a surd. This means that the solutions to this equation are irrational. This is why we could not solve the equation by factorising.

 

Summary

For a quadratic equation of the form

$Ax^2+Bx+C=0$Ax2+Bx+C=0

The solutions are given by the quadratic formula:

$x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}$x=B±B24AC2A

Practice questions

Question 1

Solve $x^2+5x+6=0$x2+5x+6=0 for $x$x by using the quadratic formula or otherwise.

Enter each solution on the same line, separated by a comma.

Question 2

Solve $4x^2+7x+3=0$4x2+7x+3=0 for $x$x by using the quadratic formula or otherwise.

Enter each solution on the same line, separated by a comma.

Question 3

Solve $10-6m+2m^2=m^2+8m+9$106m+2m2=m2+8m+9 for $m$m by using the quadratic formula or otherwise.

Enter each solution as a surd on the same line, separated by a comma.

Outcomes

VCMNA341

Solve simple quadratic equations using a range of strategies.

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