Math / Algebra

QuestionResolva essas funções encontre as raízes da função do estudo da concavidade e o vértice da parábola 10:49 letra C) $2 x^{2}+5 x-3=0 \quad \Delta=b^{2}-4 a c$ 13:31

Studdy Solution

STEP 1

1. We are given a quadratic equation of the form $2x^2 + 5x - 3 = 0$ .
2. We need to find the roots of the quadratic equation using the quadratic formula.
3. We need to study the concavity of the parabola represented by the quadratic equation.
4. We need to find the vertex of the parabola.

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation.
2. Calculate the discriminant $\Delta = b^2 - 4ac$ .
3. Use the quadratic formula to find the roots of the equation.
4. Determine the concavity of the parabola.
5. Calculate the vertex of the parabola.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $2x^2 + 5x - 3 = 0$ .
Here, $a = 2$ , $b = 5$ , and $c = -3$ .

STEP 4

Calculate the discriminant $\Delta = b^2 - 4ac$ .
$\Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49$

STEP 5

Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ to find the roots of the equation.
$x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4}$

STEP 6

Calculate the two roots of the equation.
$x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$ $x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$

STEP 7

Determine the concavity of the parabola by examining the coefficient $a$ .
Since $a = 2$ and $a > 0$ , the parabola opens upwards and is concave upwards.

STEP 8

Calculate the vertex of the parabola using the vertex formula $x = -\frac{b}{2a}$ .
$x = -\frac{5}{2 \cdot 2} = -\frac{5}{4}$

STEP 9

Find the $y$ -coordinate of the vertex by substituting $x = -\frac{5}{4}$ back into the quadratic equation.
$y = 2 \left(-\frac{5}{4}\right)^2 + 5 \left(-\frac{5}{4}\right) - 3$ $y = 2 \left(\frac{25}{16}\right) - \frac{25}{4} - 3$ $y = \frac{50}{16} - \frac{100}{16} - \frac{48}{16}$ $y = \frac{50 - 100 - 48}{16}$ $y = \frac{-98}{16} = -\frac{49}{8}$

STEP 10

The vertex of the parabola is at $\left(-\frac{5}{4}, -\frac{49}{8}\right)$ .
Solution: The roots of the quadratic equation $2x^2 + 5x - 3 = 0$ are $\frac{1}{2}$ and $-3$ . The parabola is concave upwards. The vertex of the parabola is $\left(-\frac{5}{4}, -\frac{49}{8}\right)$ .

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QuestionResolva essas funções encontre as raízes da função do estudo da concavidade e o vértice da parábola 10:49 letra C) $2 x^{2}+5 x-3=0 \quad \Delta=b^{2}-4 a c$ 13:31

Studdy Solution

STEP 1

1. We are given a quadratic equation of the form $2x^2 + 5x - 3 = 0$ .
2. We need to find the roots of the quadratic equation using the quadratic formula.
3. We need to study the concavity of the parabola represented by the quadratic equation.
4. We need to find the vertex of the parabola.

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation.
2. Calculate the discriminant $\Delta = b^2 - 4ac$ .
3. Use the quadratic formula to find the roots of the equation.
4. Determine the concavity of the parabola.
5. Calculate the vertex of the parabola.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $2x^2 + 5x - 3 = 0$ .
Here, $a = 2$ , $b = 5$ , and $c = -3$ .

STEP 4

Calculate the discriminant $\Delta = b^2 - 4ac$ .
$\Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49$

STEP 5

Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ to find the roots of the equation.
$x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4}$

STEP 6

Calculate the two roots of the equation.
$x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$ $x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$

STEP 7

Determine the concavity of the parabola by examining the coefficient $a$ .
Since $a = 2$ and $a > 0$ , the parabola opens upwards and is concave upwards.

STEP 8

Calculate the vertex of the parabola using the vertex formula $x = -\frac{b}{2a}$ .
$x = -\frac{5}{2 \cdot 2} = -\frac{5}{4}$

STEP 9

STEP 10

The vertex of the parabola is at $\left(-\frac{5}{4}, -\frac{49}{8}\right)$ .
Solution: The roots of the quadratic equation $2x^2 + 5x - 3 = 0$ are $\frac{1}{2}$ and $-3$ . The parabola is concave upwards. The vertex of the parabola is $\left(-\frac{5}{4}, -\frac{49}{8}\right)$ .

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QuestionResolva essas funções encontre as raízes da função do estudo da concavidade e o vértice da parábola 10:49 letra C) 2x2+5x−3=0Δ=b2−4ac2 x^{2}+5 x-3=0 \quad \Delta=b^{2}-4 a c2x2+5x−3=0Δ=b2−4ac 13:31

Studdy Solution

STEP 1

STEP 2

1. Identify the coefficients aaa, bbb, and ccc from the quadratic equation.2. Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.3. Use the quadratic formula to find the roots of the equation.4. Determine the concavity of the parabola.5. Calculate the vertex of the parabola.

STEP 3

Identify the coefficients aaa, bbb, and ccc in the quadratic equation 2x2+5x−3=02x^2 + 5x - 3 = 02x2+5x−3=0.Here, a=2a = 2a=2, b=5b = 5b=5, and c=−3c = -3c=−3.

STEP 4

Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.Δ=52−4(2)(−3)=25+24=49 \Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49 Δ=52−4(2)(−3)=25+24=49

STEP 5

Use the quadratic formula x=−b±Δ2ax = \frac{-b \pm \sqrt{\Delta}}{2a}x=2a−b±Δ​​ to find the roots of the equation.x=−5±492⋅2=−5±74 x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4} x=2⋅2−5±49​​=4−5±7​

STEP 6

Calculate the two roots of the equation.x1=−5+74=24=12 x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2} x1​=4−5+7​=42​=21​ x2=−5−74=−124=−3 x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 x2​=4−5−7​=4−12​=−3

STEP 7

Determine the concavity of the parabola by examining the coefficient aaa.Since a=2a = 2a=2 and a>0a > 0a>0, the parabola opens upwards and is concave upwards.

STEP 8

Calculate the vertex of the parabola using the vertex formula x=−b2ax = -\frac{b}{2a}x=−2ab​.x=−52⋅2=−54 x = -\frac{5}{2 \cdot 2} = -\frac{5}{4} x=−2⋅25​=−45​

STEP 9

STEP 10

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QuestionResolva essas funções encontre as raízes da função do estudo da concavidade e o vértice da parábola 10:49 letra C) $2 x^{2}+5 x-3=0 \quad \Delta=b^{2}-4 a c$ 13:31

1. Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation.
2. Calculate the discriminant $\Delta = b^2 - 4ac$ .
3. Use the quadratic formula to find the roots of the equation.
4. Determine the concavity of the parabola.
5. Calculate the vertex of the parabola.

Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $2x^2 + 5x - 3 = 0$ .
Here, $a = 2$ , $b = 5$ , and $c = -3$ .

Calculate the discriminant $\Delta = b^2 - 4ac$ .
$\Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49$

Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ to find the roots of the equation.
$x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4}$

Calculate the two roots of the equation.
$x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$ $x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$

Determine the concavity of the parabola by examining the coefficient $a$ .
Since $a = 2$ and $a > 0$ , the parabola opens upwards and is concave upwards.

Calculate the vertex of the parabola using the vertex formula $x = -\frac{b}{2a}$ .
$x = -\frac{5}{2 \cdot 2} = -\frac{5}{4}$