No Real Solutions When Completing the Square
Grade 9 students in California Reveal Math Algebra 1 learn that completing the square can reveal a quadratic equation has no real solutions when the right-hand side becomes negative. After completing the square to get (x+h)²=d: if d>0 there are two real solutions; if d=0 there is exactly one; if d<0 there are no real solutions because no real number squared equals a negative. For example, x²+4x+7=0 completes to (x+2)²=-3, and since -3<0 there are no real solutions. Students learn to check the sign of d before taking square roots.
Key Concepts
After completing the square, if the equation takes the form:.
$$(x + h)^2 = d$$.
Common Questions
When does completing the square reveal no real solutions?
After completing the square, if the equation takes the form (x+h)²=d where d<0, there are no real solutions because no real number squared can equal a negative number.
How do you solve x²+4x+7=0 by completing the square?
Isolate: x²+4x=-7. Complete the square by adding 4: x²+4x+4=-7+4. Factor: (x+2)²=-3. Since -3<0, there are no real solutions.
How do you solve 2x²-4x+6=0 by completing the square?
Divide by 2: x²-2x+3=0. Isolate: x²-2x=-3. Complete: x²-2x+1=-3+1. Factor: (x-1)²=-2. Since -2<0, there are no real solutions.
What should you do when you get a negative value after completing the square?
Stop and state that there are no real solutions. Do not proceed to take the square root of a negative number — that step is only valid when d≥0.
How do you know when real solutions do exist after completing the square?
If d>0 after completing the square, apply the Square Root Property: x=-h±sqrt(d). For example, (x+2)²=3 gives x=-2±sqrt(3), which has two real solutions.
Which unit covers completing the square in Algebra 1?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.