Quadratic Equations

The One Thing

A quadratic equation has the standard form $a{x}^2+bx+c=0$, where $a\ne 0$. It can be solved by methods such as factorisation or the quadratic formula, while the discriminant $D=b^2-4ac$ determines whether its roots are distinct real, equal real, or non-real.

Definitions and Results

• Quadratic equation: An equation of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are real numbers and $a\ne 0$. If $a=0$, the equation is linear rather than quadratic.
• Standard form: $a{x}^2+bx+c=0$. This form identifies the coefficients $a$, $b$, and $c$.
• Root or zero: A value of $x$ that satisfies the quadratic equation and makes the expression equal to zero.
• Factorisation: Writing a quadratic expression as a product of simpler expressions, such as $(x+p)(x+q)$.
• Zero-product property: If a product is zero, at least one factor must be zero. Thus, if $(x-p)(x-q)=0$, then $x=p$ or $x=q$.
• Discriminant: $D=b^2-4ac$. It determines the nature and number of real roots.
• Distinct real roots: If $D>0$, there are two different real roots.
• Equal real roots: If $D=0$, there is one repeated real root, with each root equal to $-\frac{b}{2a}$.
• No real roots: If $D<0$, the equation has no real-valued roots.
• Quadratic formula: For $a{x}^2+bx+c=0$,

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Sum of roots: If the roots are $\alpha$ and $\beta$, then

$$\alpha +\beta =-\frac{b}{a}.$$

• Product of roots: If the roots are $\alpha$ and $\beta$, then

$$\alpha \beta =\frac{c}{a}.$$

• Equation formed from given roots: A quadratic with roots $\alpha$ and $\beta$ is

$$x^2-(\alpha +\beta )x+\alpha \beta =0.$$

• Factorisation pattern: For

$$x^2+(p+q)x+pq=0,$$ factorisation gives $$(x+p)(x+q)=0.$$

Worked Methods

Factorisation of a monic quadratic

• Write the equation in standard form.
• Find two numbers whose product is the constant term and whose sum is the coefficient of $x$.
• Write the quadratic as a product of two linear factors.
• Apply the zero-product property.
• Solve each resulting linear equation.

For $$x^2-5x+6=0,$$ the numbers $-2$ and $-3$ have product $6$ and sum $-5$. Therefore, $$x^2-5x+6=(x-2)(x-3),$$ so $$(x-2)(x-3)=0.$$ Hence, $$x=2\text{or}x=3.$$

Splitting the middle term

For a quadratic expression $$a{x}^2+bx+c,$$ find two numbers whose product is $ac$ and whose sum is $b$. Use these numbers to split the middle term, then factor by grouping. If $$a{x}^2+bx+c=(lx+m)(nx+r),$$ set each factor equal to zero: $$lx+m=0\text{or}nx+r=0.$$

Using the discriminant

• Identify $a$, $b$, and $c$ from the standard form.
• Calculate

$$D=b^2-4ac.$$

• Interpret the result:

- $D>0$: two distinct real roots; - $D=0$: two equal real roots; - $D<0$: no real roots.

For $$2x^2+4x+2=0,$$ $$D=4^2-4(2)(2)=0.$$ Therefore, the roots are real and equal. Each repeated root is $$-\frac{b}{2a}=-\frac{4}{2(2)}=-1.$$

Using the quadratic formula

• Rewrite the equation as $a{x}^2+bx+c=0$.
• Identify $a$, $b$, and $c$, including their signs.
• Substitute into

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Simplify both values arising from the $+$ and $-$ signs.
• Verify the roots by substitution into the original equation.

Using the sum and product of roots

If the roots are $\alpha$ and $\beta$, use $$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$ These relationships provide a check on calculated roots. Conversely, an equation with roots $\alpha$ and $\beta$ is formed as $$x^2-(\alpha +\beta )x+\alpha \beta =0.$$

Where It Goes Wrong

• Treating an equation with $a=0$ as quadratic; when $a=0$, the equation is linear.
• Choosing factors without checking both their product and their sum, particularly when splitting the middle term.
• Failing to apply the zero-product property after factorisation by setting each factor equal to zero.
• Misidentifying $a$, $b$, or $c$ in the quadratic formula, especially when coefficients have negative signs.
• Forgetting that $D=0$ gives equal real roots, each equal to $-\frac{b}{2a}$, rather than two distinct roots.
• Neglecting to verify the obtained roots by substitution into the original equation.

What Gets Asked

• Define a quadratic equation and identify its coefficients from standard form.
• Factorise a quadratic and solve it using the zero-product property.
• Solve $x^2-5x+6=0$, obtaining roots $2$ and $3$.
• Split the middle term using numbers whose product is $ac$ and sum is $b$.
• Calculate the discriminant and state whether the roots are distinct real, equal real, or non-real.
• For $2x^2+4x+2=0$, calculate $D=4^2-4(2)(2)=0$ and determine the repeated root.
• Solve a quadratic using

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• State or use

$$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$

• Form a quadratic equation from specified roots $\alpha$ and $\beta$.
• Check a solution by substituting the roots into the original equation.