"Discriminant of Roots of Quadratic Equations" PPT Courseware 2

Description

"Discriminant of Roots of Quadratic Equations" PPT Courseware 2

Knowledge review

1. What is the formula for finding the root of a quadratic equation?

Generally speaking, for the quadratic equation ax²+bx+c=0 (a≠0), when b²-4ac≥0, its root is

x=-b±√b²-4ac/2a

2. What are the general steps for solving quadratic equations of one variable using the formula method?

To use the formula method to solve a quadratic equation of one variable, you must first convert it into a general form.

Then determine the values ​​of a, b, c, and then find the value of b2-4ac,

When b²-4ac≥0, then substitute it into the formula to solve;

When b²-4ac<0, the equation has no real solution (root)

Summary

From this we can find that the roots of the quadratic equation ax²+bx+c=0 (a≠0) can be determined by b²-4ac

When b²-4ac>0, the equation has two unequal real roots

When b²-4ac=0, the equation has two equal real roots

When b²-4ac<0, the equation has no real roots

We call b²-4ac the discriminant of the roots of the quadratic equation ax²+bx+c=0 (a≠0).

If we know the roots of a quadratic equation, can we get the sign of the discriminant value?

When a quadratic equation has two unequal real roots, b²-4ac>0

When a quadratic equation has two equal real roots, b²-4ac = 0

When the quadratic equation has no real roots, b²-4ac<0

Concept consolidation

1. The discriminant b2-4ac=-8 of equation 3x²+2=4x,

So the case for the roots of an equation is that the equation has no real roots.

2. Among the following equations, the equation without real roots is ( )

A.x²=9 B.4x²=3(4x-1)

C.x(x+1)=1 D.2y²+6y+7=0

3. The equation ax2+bx+c=0(a≠0) has real roots, then the formula that always holds true is ( )

A.b²-4ac＞0 B.b²-4ac＜0

C.b²-4ac≤0 D.b²-4ac≥0

Typical examples

Example 1 Without solving the equation, determine the roots of the following equations:

(1)-x²+2√6x-6=0

(2)x²+4x=2

(3)4x²+1=-3x

(4)x²-2mx+4(m-1)=0

Solution (1)∵b²-4ac=24-4×(-1)×(-6)=0

∴This equation has two equal real roots

(2) Move the term to get x²+4x-2=0

∵b²-4ac=16-4×1×(-2)=16-(-8)

=16+8=24>0

∴This equation has two unequal real roots

practice

1. Without solving the equation, determine the roots of the equation:

(1)x²+3x-1=0;

(2)x²-6x+9=0;

(3)2y²-3y+4=0

(4)x²+5=2√5x

2. When k takes on what value, the equation x²-kx+4=0 has two equal real roots? Find the roots of this equation.

3. It is known that a, b, and c are the three sides of the triangle respectively, then the root of the quadratic equation of x (a+b)x²+2cx+(a+b)=0 is ( )

A. There are no real roots

B. There may be and only one real root

C. There are two equal real roots

D. There are two unequal real roots.

In conclusion

What is the relationship between the roots of a quadratic equation and its coefficients?

b²-4ac is called the discriminant of the roots of a quadratic equation of one variable. The root discriminant can be used to determine the roots of a quadratic equation without solving the equation; conversely, the symbol of b²-4ac can also be known from the roots of the equation, and then the unknown letters in the equation can be obtained. value situation.

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"Discriminant of Roots of Quadratic Equations" PPT courseware:

"Discriminant of Roots of Quadratic Equations" PPT courseware A. Introduced from solving equations: Solving equations: ①x+x-1=0 b-4ac=1+4=5 ②x-4x+4=0 b-4ac=16-16=0 ③2x+3x+4=0 b-4ac =9-32<0 This equation has no real roots. It can be judged from the value of b-4ac..

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Update Time: 2024-07-11

This template belongs to Mathematics courseware Qingdao Edition Ninth Grade Mathematics Volume 1 industry PPT template

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