Here’s a quadratic equation in standard form: ax2{\displaystyle ax^{2}}+bx+c=0{\displaystyle +bx+c=0}[2] X Research source Here are 2 examples to demonstrate: x2−3x+1=0{\displaystyle x^{2}-3x+1=0}This equation is ready to solve because it equals 0{\displaystyle 0}. −3x2+6x=−5{\displaystyle -3x^{2}+6x=-5}This equation is not ready to solve just yet. We need to convert it first.
If an equation looks like this: −3x2+6x=−5{\displaystyle -3x^{2}+6x=-5} Move the −5{\displaystyle -5} to the left side of the equal sign and put 0{\displaystyle 0} on the right side of the equal sign. Remember: numerals change from +{\displaystyle +} to −{\displaystyle -} (or vice versa) when you move them to the other side of the equal sign. Our converted equation: −3x2+6x+5=0{\displaystyle -3x^{2}+6x+5=0} If an equation looks like this: x2=3x−1{\displaystyle x^{2}=3x-1} Move all the terms to left side of the equal sign. Our converted equation: x2−3x+1=0{\displaystyle x^{2}-3x+1=0} If an equation looks like this: 2(w2−2w)=5{\displaystyle 2(w^{2}-2w)=5} Undo the brackets to expand and move 5 to the left of the equal sign. Our converted equation: 2w2−4w−5=0{\displaystyle 2w^{2}-4w-5=0}
Move the −5{\displaystyle -5} to the left side of the equal sign and put 0{\displaystyle 0} on the right side of the equal sign. Remember: numerals change from +{\displaystyle +} to −{\displaystyle -} (or vice versa) when you move them to the other side of the equal sign. Our converted equation: −3x2+6x+5=0{\displaystyle -3x^{2}+6x+5=0}
The coefficients in our equation: a=−3{\displaystyle a=-3} b=6{\displaystyle b=6} c=5{\displaystyle c=5}
Remember, the quadratic formula looks like this:x=−b{\displaystyle x=-b}± √(b2−4ac){\displaystyle b^{2}-4ac)}/2a{\displaystyle /2a} Our coefficients: a=−3{\displaystyle a=-3}, b=6{\displaystyle b=6}, and c=5{\displaystyle c=5} Our equation after inserting the coefficients:x=−6{\displaystyle x=-6}± √(62−(4)(−3)(5){\displaystyle 6^{2}-(4)(-3)(5)}/2(−3){\displaystyle /2(-3)}
62=36{\displaystyle 6^{2}=36} 2{\displaystyle 2} x −3=−6{\displaystyle -3=-6} −3{\displaystyle -3} x 5=−15{\displaystyle 5=-15} −15{\displaystyle -15} x −4{\displaystyle -4} = 60{\displaystyle 60} You end up with: x=−6{\displaystyle x=-6}± √(36+60){\displaystyle (36+60)}/−6{\displaystyle /-6} Then, simplify once more: x=−6{\displaystyle x=-6}± √96{\displaystyle 96}/−6{\displaystyle /-6}
Here’s the prime factorization of 96: 2 x 2 x 2 x 2 x 2 x 3 = 96. Group the pairs: (2 x 2) (2 x 2). There are four 2s, so 4 goes outside the radical sign. Multiply what’s left: (2 x 3) = 6. This goes inside the radical sign. So √96{\displaystyle 96} simplified = 4√6{\displaystyle 6} Putting it all together: x=−6{\displaystyle x=-6}± 4√6{\displaystyle 6}/−6{\displaystyle /-6}
-6 ÷ 2 = -3 4 ÷ 2 = 2 6 ÷ 2 = 3 The reduced equation: x=−3{\displaystyle x=-3}± 2√6{\displaystyle 6}/−3{\displaystyle /-3} or x=3{\displaystyle x=3}± 2√6{\displaystyle 6}/3{\displaystyle /3}(both answers are correct because of the ± sign) These are your final answers. [9] X Research source
These are your final answers. [9] X Research source
Sing these lyrics to the tune of Pop Goes the Weasel:X is equal to negative BPlus or minus the square rootOf B-squared minus four A CAll over two A If songs aren’t your thing, try memorizing this story instead:A negative boy was thinking yes or no about going to a party. At the party, he talked to a square boy but not to the 4 awesome cats. It was all over at 2 am. [10] X Research source
