quadratic formula
Solve any quadratic equation ax²+bx+c=0 in Grade 10 algebra using the quadratic formula x=(-b±√(b²-4ac))/(2a), and use the discriminant b²-4ac to determine the number and type of roots.
Key Concepts
Property To find the roots of any quadratic equation in the form $ax^2 + bx + c = 0$, you can use the quadratic formula: $$x = \frac{ b \pm \sqrt{b^2 4ac}}{2a}$$.
To solve $2x^2 + 6x 3 = 0$, identify $a=2, b=6, c= 3$. $x = \frac{ (6) \pm \sqrt{(6)^2 4(2)( 3)}}{2(2)} = \frac{ 6 \pm \sqrt{36 + 24}}{4}$. $x = \frac{ 6 \pm \sqrt{60}}{4} = \frac{ 6 \pm 2\sqrt{15}}{4} = \frac{ 3 \pm \sqrt{15}}{2}$.
Think of the quadratic formula as the ultimate cheat code for solving these types of equations. Instead of wrestling with factoring or completing the square every single time, you can just plug in your $a$, $b$, and $c$ values. It’s a powerful shortcut that works for any quadratic equation you encounter.
Common Questions
What is the quadratic formula?
For ax²+bx+c=0, x = (-b ± √(b²-4ac)) / (2a). It always gives the solution(s) for any quadratic equation.
What does the discriminant b²-4ac tell you about the roots?
If b²-4ac>0: two distinct real roots. If b²-4ac=0: one repeated real root. If b²-4ac<0: two complex conjugate roots (no real solutions).
Solve 2x²-3x-5=0 using the quadratic formula.
a=2, b=-3, c=-5. x=(3±√(9+40))/4=(3±7)/4. Solutions: x=10/4=5/2 and x=-4/4=-1.