二次方程式在我们现实生活中几乎无处不在。例如,甚至可以将设计运动场的问题表述为二次方程。当许多情况产生二次方程式时,就会激起人们对寻找其解的真正兴趣。假设Q(x)= 0是一个二次方程。二次方程的解表示满足该方程的点Q(x)=0。该解也称为二次方程的根/零。让我们看一下解决二次方程式的一些方法。
二次方程
二次方程是二次多项式。它的一般形式由下式给出:
斧2 +斧+ c = 0
a,b和c是实数,而a≠0。其形状是一个抛物线,根据“ a”的值向上或向下开口。
它的解决方案是满足方程式的点。有几种方法可以找到二次方程的解,如下所示:
- 保理
- 完成广场
- 二次公式
保理
我们尝试将方程式分解出来,以得到两个项的乘积形式的方程式。然后,将这两项等同为零,便得到了根。
The following steps must be used for finding the roots with factorization:
- All the terms must be on one side of the equation, either LHS or RHS leaving zero on the other side.
- Factorize the equation
- Set the factors equal to zero to find the roots one by one.
让我们使用以下示例更详细地研究此方法:
问题1:分解以下等式并找到其根:2x 2 – x – 1 = 0
解决方案:
2x2 – x – 1 = 0
⇒ 2x2 -2x + x – 1 = 0
⇒ 2x(x – 1) + 1(x – 1) = 0
⇒ (2x + 1) (x – 1) = 0
For this equation two be zero, either one of these or both of these terms should be zero.
So, we can find out roots by equating these terms with zero.
2x + 1 = 0
x =
x – 1 = 0
⇒ x = 1
So, we get two roots in the equation.
x = 1 and
问题2:分解以下等式并找到其根:x 2 + x – 12 = 0
解决方案:
x2 + x – 12 = 0
⇒ x2 + 4x – 3x – 12 = 0
⇒ x(x + 4) -3(x + 4) = 0
⇒ (x – 3) (x + 4) = 0
Equating both of these terms with zero.
x – 3 = 0 and x – 4 = 0
x = 3 and 4
完成广场
我们尝试将方程式设为完整正方形,例如:(x – a) 2 – b 2 = 0。
Steps for finding out roots by completing the square method:
Step 1: Bring the equation in the form ax2 + bx = -c.
Step 2: We need to make sure that a = 1 (if a≠1, multiply through the equation by before going to next step.)
Step 3: Use the value of b from this new equation and to both sides of the equation to form a perfect square on the left side of the equation.
Step 4: Find the square root of the both sides of the equation.
Step 5: Solve the result to get the roots.
让我们来看一些有关此的示例,
问题1:通过完成平方法,找到以下方程式的根。
4x 2 + 12x + 9 = 0。
解决方案:
4x2 + 12x + 9 = 0
⇒ (2x)2 + 2(3)(2)x + 32 = 0
We can see that this equation is a perfect square,
⇒ (2x + 3)2 = 0
To find out the zeros in this equation,
2x + 3 = 0
x =
This equation has repeating root, which is x =
上面的问题的方程式是一个完美的平方,但并非每次都这样。在这些情况下,我们将使用上述步骤以上述形式给出方程式。
问题2:通过完成平方的方法来找到方程式的根。
9x 2 + 24x + 3 = 0
解决方案:
9x2 + 24x + 3 = 0
This equation can be re-written as,
⇒ 9x2 + 24x + 16 – 13 = 0
⇒ (3x)2 + 24x + 42 -13 =0
⇒ (3x + 4)2 -13 = 0
⇒ (3x + 4)2 -(√13)2 = 0
⇒ (3x + 4)2 = (√13)2
Taking square root of the both sides of the equation.
3x + 4 = √13 or 3x + 4 = -√13
We get our roots by solving these two equations,
3x + 4 = √13
x =
Similarly,
3x + 4 = – √13
x =
二次公式
所有二次方程都可以使用二次方程求解。
For an equation of the form,
ax2 + bx + c = 0,
Where a, b and c are real numbers and a ≠ 0.
The roots of this equation are given by,
x =
Given that b2 – 4ac is greater than or equal to zero.
问题1:使用二次公式找出方程的根,
4x 2 + 10x + 3 = 0
回答:
4x2 + 10x + 3 = 0
Using Quadratic Formula to solve this,
a = 4, b = 10 and c = 3
Before plugging in the values, we need to check for the discriminator
b2 – 4ac
⇒ 102 – 4(4)(3)
⇒ 100 – 48
⇒ 52
This is greater than zero, So now we can apply the quadratic formula.
Plugging the values into quadratic equation,
问题2:使用二次公式找出方程的根,
5x 2 + 9x + 4 = 0
解决方案:
5x2 + 9x + 4 = 0
Using Quadratic Formula,
a = 5, b = 9 and c = 4.
Before plugging in the values, we need to check for the discriminator
b2 – 4ac
⇒ 92 – 4(5)(4)
⇒ 81 – 80
⇒ 1
This is greater than zero, So the quadratic formula can be applied. Plugging in the values in the formula,