(3.5) Solving Quadratic Equations
Equations in the form $ax^2+bx+c=0$ where $a\ne 0$ are called quadratic equations.
In this section, we will look at strategies to solve them.
Null Factor Law
Fill in the blanks.
$6\times \square =0$
$\left(-3\right)\times \square =0$
$a\times \square =0$
$\square \times 7=0$
$\square \times \left(-5\right)=0$
$\square \times b=0$
Fill in the blank. If $a\times b=0$, then $a=\square$ or $b=\square$ or both.
Fill in the blanks.
$6\times \left(\square-5\right) =0$
$\left(-3\right) \left(\square +4\right) =0$
$a\times \left(\square -p\right) =0$
$\left(\square +9\right) \times 7=0$
$\left(\square -\frac{1}{2}\right) \left(-5\right)=0$
$\left(\square -q\right) \times b=0$
Solve for $x$. If there are multiple answers, separate them with commas.
$\left(x-2\right)\left(x-1\right)=0\qquad \qquad x=$
$\left(x-7\right)\left(x+7\right)=0\qquad \qquad x=$
$\left(x-4\right)^2=0\qquad \qquad x=$
$x\left(x+8\right)=0\qquad \qquad x=$
$3\left(x-5\right)\left(x+4\right)=0\qquad \qquad x=$
$-2\left(x+6\right)^2=0\qquad \qquad x=$
Solving by Factorizing
Fully factorize. $\qquad x^2-3x+2=$
Solve for $x$.$\qquad x^2-3x+2=0\qquad \qquad x=$
Fully factorize. $\qquad x^2-49=$
Solve for $x$.$\qquad x^2-49=0\qquad \qquad x=$
Fully factorize. $\qquad x^2-8x+16=$
Solve for $x$.$\qquad x^2-8x+16=0\qquad \qquad x=$
Fully factorize. $\qquad x^2+8x=$
Solve for $x$.$\qquad x^2+8x=0\qquad \qquad x=$
Fully factorize. $\qquad 3x^2-3x-60=$
Solve for $x$.$\qquad 3x^2-3x-60=0\qquad \qquad x=$
Fully factorize. $\qquad -2x^2-24x-72=$
Solve for $x$.$\qquad -2x^2-24x-72=0\qquad \qquad x=$
Solve for $x$. If there are multiple answers, separate them with commas.
$x^2-3x=-2\qquad \qquad x=$
$x^2=49\qquad \qquad x=$
$x^2+16=8x\qquad \qquad x=$
$8x=-x^2\qquad \qquad x=$
$3\left(x^2-20\right)=3x\qquad \qquad x=$
$-72=2x\left(x+12\right)\qquad \qquad x=$
Solve for $𝑥$. If there are multiple answers, separate them with commas.
$x^2+7x=0\quad x=$
$x^2+8x=0\quad x=$
Solve for $𝑥$. If there are multiple answers, separate them with commas.
$x^2=-7x\quad x=$
$x^2=-8x\quad x=$
Solve for $𝑥$. If there are multiple answers, separate them with commas.
$4x^2-3x=0\quad x=$
$5x^2+7x=0\quad x=$
$8x^2-3x=0\quad x=$
$-8x^2+7x=0\quad x=$
Solve for $𝑥$. If there are multiple answers, separate them with commas.
$-4x^2=-5x\quad x=$
$5x^2=4x\quad x=$
$7x^2=-3x\quad x=$
$9x^2=4x\quad x=$