Solving Quadratic Equations Flashcards

All 6 cards in this deck

What is the quadratic formula for solving $ax^2+bx+c=0$?

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

If a product of two factors equals zero, what must be true?

e.g. $(x+3)(2x-1)=0$

At least one factor is zero.
e.g. $x+3=0$ or $2x-1=0$, so $x=-3$ or $x=\frac{1}{2}$

What are the steps to solve a quadratic equation by factorising?

e.g. $x^2 + x - 6 = 0$

1. Make sure one side is $0$ — it already is
2. Factorise: $(x+3)(x-2)=0$
3. Set each bracket to zero: $x=-3$ or $x=2$

What are the steps to solve a quadratic given in completed square form?

e.g. $(x-1)^2 - 5 = 0$

1. Isolate the squared bracket: $(x-1)^2 = 5$
2. Take the square root of both sides, with $\pm$: $x-1 = \pm\sqrt{5}$
3. Solve for $x$: $x = 1 \pm \sqrt{5}$

What are the steps to solve an equation that becomes a quadratic after expanding?

e.g. $x(x+2)=15$

1. Expand: $x^2+2x=15$
2. Collect all terms on one side: $x^2+2x-15=0$
3. Factorise and solve: $(x+5)(x-3)=0$, so $x=-5$ or $x=3$

True or false? To solve $x^3 - 3x^2 = 0$ you can divide both sides by $x^2$.

False. You lose the solution $x=0$ — factorise to $x^2(x-3)=0$ giving $x=0$ or $x=3$.