An equation is quadratic only once it's simplified: expand both sides fully and collect like terms first, then check the degree. Two examples with different outcomes:
(x+1)2=2(x−3) expands to x2+2x+1=2x−6, which simplifies to x2+7=0, degree 2, so this is a quadratic equation.
(x−2)(x+1)=(x−1)(x+3) expands to x2−x−2=x2+2x−3, and the x2 terms cancel, leaving −3x+1=0, only degree 1, so this is not a quadratic equation, despite looking like one before expanding.
Turning a word problem into a quadratic equation
The product of two consecutive positive integers is 306. Represent this as a quadratic equation.
Solution: Let the first integer be x, so the next consecutive integer is x+1. Their product is 306:
x(x+1)=306⇒x2+x−306=0
Solving by factorization
Solve 2x2+x−6=0.
Solution: Split the middle term into two parts whose product matches 2×(−6)=−12 and whose sum is 1: that's 4 and −3.
2x2+4x−3x−6=0⇒2x(x+2)−3(x+2)=0⇒(x+2)(2x−3)=0
So x=−2 or x=23.
A word problem solved end to end
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Solution: Let the base be x cm, so the altitude is x−7 cm. By the Pythagorean theorem:
132=x2+(x−7)2⇒169=2x2−14x+49⇒x2−7x−60=0x2−12x+5x−60=0⇒(x−12)(x+5)=0⇒x=12 or x=−5
A side length can't be negative, so x=12: the base is 12 cm and the altitude is 12−7=5 cm, a 5-12-13 right triangle, and 52+122=132 checks out.
Solving by completing the square
Solve 2x2−7x+3=0 by completing the square.
Solution: Divide by 2, then move the constant to the right:
x2−27x=−23
Add (47)2=1649 to both sides to complete the square on the left:
(x−47)2=16−24+49=1625⇒x−47=±45
So x=412=3 or x=42=21.
The nature of the roots
The discriminant b2−4ac tells you what kind of roots to expect before you even solve: positive means two distinct real roots, zero means one repeated real root, and negative means no real roots at all.
For what value(s) of k does 2x2+kx+3=0 have two equal roots?
Solution: Equal roots require b2−4ac=0:
k2−4(2)(3)=0⇒k2=24⇒k=±26
Continue learning
Every quadratic equation here has real roots because its discriminant is positive or zero. When the discriminant is negative instead, the roots become a conjugate pair of complex numbers.