Number of Solutions in a Linear-Quadratic System
Grade 9 students in California Reveal Math Algebra 1 learn to determine the number of solutions in a linear-quadratic system using the discriminant. A line and parabola can intersect at 0, 1, or 2 points. To find out algebraically, substitute the linear expression into the quadratic to produce a single equation ax²+bx+c=0, then evaluate the discriminant b²-4ac: if positive there are 2 solutions, if zero there is 1 solution (tangent), and if negative there are 0 solutions. For example, y=x+3 and y=x²-1 gives discriminant 17>0, so 2 solutions; y=x and y=x²+3 gives discriminant -11<0, so no solutions.
Key Concepts
A system of one linear equation and one quadratic equation can have 0, 1, or 2 real solutions , depending on how many times the line intersects the parabola.
To determine the number of solutions algebraically, substitute the linear expression into the quadratic equation to obtain a single quadratic equation of the form:.
Common Questions
How many solutions can a linear-quadratic system have?
A system of one linear equation and one quadratic equation can have 0, 1, or 2 real solutions, depending on how many times the line intersects the parabola.
How do you use the discriminant to count solutions in a linear-quadratic system?
Substitute the linear expression into the quadratic to get ax²+bx+c=0. Then compute the discriminant b²-4ac. If it is positive: 2 solutions; if zero: 1 solution; if negative: 0 solutions.
What does it mean when the discriminant equals zero in this system?
A discriminant of zero means the line is tangent to the parabola — it touches it at exactly one point without crossing through it.
Can you show a system with two solutions?
For y=x+3 and y=x²-1: substituting gives x²-x-4=0. The discriminant is (-1)²-4(1)(-4)=1+16=17>0, so there are 2 real solutions.
Can you show a system with no solutions?
For y=x and y=x²+3: substituting gives x²-x+3=0. The discriminant is (-1)²-4(1)(3)=1-12=-11<0, so there are no real solutions and the line does not intersect the parabola.
Which unit covers linear-quadratic systems in Algebra 1?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.