Quadratic Equations Flashcards

All 21 cards in this deck

If $(x+a)(x+b)=0$, what can you say?

e.g. $(x+2)(x-7)=0$

One bracket must equal $0$: $x=-a$ or $x=-b$.

e.g. $x=-2$ or $x=7$

What are the steps to solve a quadratic equation by factorising when $a=1$?

e.g. solve $x^2+5x+6=0$

1. Find two numbers with product $c$ and sum $b$: $2$ and $3$
2. Write the brackets: $(x+2)(x+3)=0$
3. Set each bracket to $0$: $x=-2$ or $x=-3$

What are the steps to solve a quadratic equation by factorising when $a\neq1$?

e.g. solve $2x^2+7x+3=0$

1. Find two numbers with product $ac=6$ and sum $b=7$: $6$ and $1$
2. Factorise into two brackets: $(2x+1)(x+3)=0$
3. Set each bracket to $0$: $x=-\frac{1}{2}$ or $x=-3$

What must you do before factorising to solve $x^2=5x+24$?

Rearrange into the form $ax^2+bx+c=0$: $x^2-5x-24=0$

True or false? The solutions of $(3k-2)(k+4)=0$ are $k=2$ and $k=-4$.

False. $3k-2=0$ gives $k=\frac{2}{3}$, so the solutions are $k=\frac{2}{3}$ and $k=-4$.

True or false? To solve $x^2=5x$ you can divide both sides by $x$ to get $x=5$.

False. Dividing by $x$ loses the solution $x=0$; factorise instead: $x(x-5)=0$, so $x=0$ or $x=5$.

State the quadratic formula.

For $ax^2+bx+c=0$, $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

In the quadratic formula, what does the $\pm$ sign tell you?

There are two solutions — one using $+\sqrt{\ }$ and one using $-\sqrt{\ }$.

True or false? In the quadratic formula only the $\pm\sqrt{b^2-4ac}$ part is divided by $2a$.

False. The whole of $-b\pm\sqrt{b^2-4ac}$ is divided by $2a$.

True or false? For $5x^2-4x-3=0$, the values to substitute are $a=5$, $b=-4$, $c=-3$.

True. The minus signs belong with the coefficients.

How many solutions does a quadratic equation have if $b^2-4ac=0$?

One (repeated) solution.

e.g. $x^2-2x+1=0$ gives only $x=1$

How do you write $x^2+bx+c$ in completed square form?

e.g. $x^2+6x+2$

$\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$

e.g. $(x+3)^2-9+2=(x+3)^2-7$

What are the steps to solve a quadratic equation by completing the square?

e.g. solve $x^2+6x+2=0$, giving exact answers

1. Write in completed square form: $(x+3)^2-7=0$
2. Rearrange and take both square roots: $x+3=\pm\sqrt{7}$
3. Solve for $x$: $x=-3\pm\sqrt{7}$

True or false? $(x+4)^2=25$ gives only the solution $x=1$.

False. Both the positive and negative square roots are needed: $x+4=\pm5$, so $x=1$ or $x=-9$.

What must you do first to complete the square when $a\neq1$?

e.g. $2x^2+8x+3=0$

Divide every term by $a$ so the coefficient of $x^2$ is $1$.

e.g. $x^2+4x+1.5=0$

What are the solutions of $(x+p)^2=q$?

e.g. $(x-2)^2=5$

$x=-p\pm\sqrt{q}$

e.g. $x=2\pm\sqrt{5}$

Which method should you use when a quadratic equation asks for an exact answer in surd form, e.g. $\frac{p\pm q\sqrt{15}}{2}$?

Completing the square, or the quadratic formula left in surd form (not rounded).

True or false? Every quadratic equation can be solved by factorising.

False. Many have no integer factorisation, so use the quadratic formula or complete the square.

A quadratic equation from a length or area problem gives one positive and one negative solution. What do you do?

Reject the negative solution (a length cannot be negative) and give only the positive value.

Where on the graph of $y=x^2+3x-3$ are the solutions of $x^2+3x-3=0$ read off?

At the $x$-coordinates of the points where the curve crosses the $x$-axis.

True or false? To find approximate solutions of $x^2-2x=4$ from the graph of $y=x^2-2x$, you read off where the curve crosses the $x$-axis.

False. You read off the $x$-values where the curve meets the line $y=4$.