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"Solving quadratic equations of one variable using formula method" PPT courseware 3
We reused the same computational steps for each specific quadratic equation;
Can you use these calculation steps for the general form of the quadratic equation ax²+bx+c=0(a≠0) to find the formula to solve x.
Use this formula to find the solution to each specific quadratic equation and achieve twice the result with half the effort.
Explore new knowledge
Solve the general form of quadratic equations of one variable using the formula method
ax²+bx+c=0（a≠0）
The root formula of the quadratic equation ax²+bx+c=0:
x=-b±√b²-4ac/2a(b²-4ac≥0)
Using this formula, we can directly obtain the solution of the equation from the values of the coefficients a, b, and c in the quadratic equation. This method of solving equations is called the formula method.
Example 1 Solve the equation: x²-7x-18=0
Solution: ∵a=1, b=-7, c=-18
∵b²-4ac=(-7)²-4×1×(-18)=121
∴x=-b±√b²-4ac/2a=-(-7)±√121/2×1=7 ± 11/2
That is: x1=9 x2=-2
Summary
General steps for solving quadratic equations of one variable using the formula method:
1. Put the equation into a general form.
2. Write the values of a, b, and c.
3. Find the value of b²-4ac.
Special note: If b²-4ac＜0, the equation has no solution
4. Substitute the root formula: x=-b±√b²-4ac/2a
5. Write the solution to the equation: x1x2
According to the sign of the value of b²-4ac, the roots of the quadratic equation ax²+bx+c=0 can be determined,
So we call b²-4ac the discriminant of the roots of the quadratic equation ax²+bx+c=0.
The roots of the quadratic equation ax²+bx+c=0(a≠0) can be determined by b²-4ac:
When △>0, the equation has two unequal real roots,
When △ = 0, there are two equal real roots,
When △<0, there are no real roots.
The equation m²x²+(2m+1)x+1=0 about x has two unequal real roots, then m_______
Variation 1: The equation m²x²+(2m+1)x+1=0 about x has two equal real roots, then m________
2: The equation m²x²+(2m+1)x+1=0 about x has no real roots, then m_______
3: The equation m²x²+(2m+1)x+1=0 about x has two real roots, then m= ________
4: The equation m²x²+(2m+1)x+1=0 about x has real roots, then m= ________
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