Photomath Is WRONG! Completing the Square

Math Professor Kyle Martin
28 Dec 202103:15

TLDRIn this video, the presenter challenges Photomath's solution to a quadratic equation, which incorrectly suggests 'no real numbers' as an answer. The presenter demonstrates the correct method of solving the equation by completing the square, revealing that the correct answer involves imaginary numbers, contrary to Photomath's initial response. The video serves as a tutorial on completing the square and a critique of relying solely on app-based solutions without understanding the underlying math.

Takeaways

  • 🔍 The video discusses an issue with Photomath's answer to a math problem involving completing the square.
  • 📱 Photomath initially provides an incorrect answer suggesting no real number solution, which is misleading.
  • 👀 The presenter demonstrates that the correct approach is to solve the equation by completing the square.
  • 🧮 The equation involves terms divisible by three, which simplifies the process.
  • ➗ The presenter moves the constant term to the other side of the equation to prepare for completing the square.
  • 🔢 The 'b' term (coefficient of x) is divided by 2 and squared to complete the square.
  • 📐 The equation is then rewritten as a binomial squared, facilitating the extraction of the square root.
  • 🚫 The square root of a negative number is identified as an imaginary number, not a real number.
  • 🔄 The presenter corrects the process by isolating x and considering both positive and negative square roots.
  • 📝 The final solution is presented as x equals negative 1 plus or minus the square root of 2, contradicting Photomath's initial answer.
  • 💡 The video serves as a tutorial on correctly completing the square and a critique of Photomath's automated solution.

Q & A

  • What issue is the speaker addressing with Photomath's answer?

    -The speaker is addressing an issue where Photomath provides an incorrect or misleading answer that suggests there are no real number solutions when in fact there are, due to the presence of imaginary numbers.

  • What mathematical method does the speaker use to solve the equation?

    -The speaker uses the method of 'completing the square' to solve the equation.

  • Why does the speaker say Photomath's initial answer is wrong?

    -The speaker says Photomath's initial answer is wrong because it incorrectly states that there are no real number solutions, when the correct answer involves complex numbers.

  • What is the first step in completing the square according to the speaker?

    -The first step in completing the square, as per the speaker, is to make the equation divisible by three by dividing each term by three.

  • How does the speaker move the constant term to the other side of the equation?

    -The speaker moves the constant term to the other side by subtracting 3 from both sides of the equation.

  • What is the 'b term' in the context of completing the square?

    -In the context of completing the square, the 'b term' refers to the coefficient of the x term in the quadratic equation.

  • What does the speaker do after identifying the b term?

    -After identifying the b term, the speaker divides it by 2 and then squares the result to complete the square.

  • Why does the speaker add 1 to both sides of the equation?

    -The speaker adds 1 to both sides of the equation to balance it after completing the square, which allows the equation to be expressed as a binomial squared.

  • How does the speaker handle the square root of a negative number?

    -The speaker handles the square root of a negative number by factoring out the negative sign and taking the square root of the positive number, which results in the imaginary unit i.

  • What is the final answer to the equation according to the speaker?

    -The final answer to the equation, according to the speaker, is x equals negative 1 plus or minus i times the square root of 2.

  • Why does the speaker believe Photomath's answer is not the first one listed?

    -The speaker believes Photomath's answer is not the first one listed because the correct answer involves complex numbers, which Photomath initially failed to recognize.

Outlines

00:00

📚 Photomath's Incorrect Answer and Completing the Square

The speaker begins by questioning the accuracy of Photomath, a popular app for solving mathematical problems. They demonstrate an instance where Photomath incorrectly suggests that no real number solution exists for an equation, indicated by a peculiar symbol. The speaker then clarifies that this is due to the presence of imaginary numbers, which Photomath incorrectly labels as 'no real numbers.' To correct this, the speaker explains the process of solving the equation by completing the square, a method not initially suggested by Photomath. They walk through the steps of dividing by three, moving terms to the other side of the equation, and then adding and subtracting the square of half the coefficient of the x term to complete the square. The final solution involves taking the square root of both sides, leading to a correct answer that includes both real and imaginary components, contrary to Photomath's initial response.

Mindmap

Keywords

💡Photomath

Photomath is a mobile application that uses optical character recognition (OCR) technology to scan and solve mathematical problems. In the context of the video, the presenter challenges the accuracy of Photomath by demonstrating an instance where it fails to provide the correct answer as the first option, which could lead to confusion for users.

💡Completing the Square

Completing the square is a mathematical method used to solve quadratic equations. It involves manipulating the equation to express it in the form of a perfect square trinomial plus a constant. The video script uses this method to solve an equation that Photomath incorrectly identified as having no real solutions, showing the correct steps to arrive at a valid solution.

💡Real Numbers

Real numbers are all the numbers that can be represented on a number line, including integers, fractions, and decimals. The video discusses how Photomath initially indicates that the equation has no solution in real numbers, which is incorrect as per the presenter's demonstration, highlighting the importance of understanding the nature of solutions in mathematics.

💡Imaginary Numbers

Imaginary numbers are a class of complex numbers that are not real, used to extend the concept of real numbers. They are represented by a real number multiplied by the imaginary unit 'i', where i^2 = -1. The video mentions that Photomath's initial answer suggests the presence of imaginary numbers, which the presenter disputes by showing that the correct solution involves real numbers.

💡Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically in the form ax^2 + bx + c = 0. The video's main theme revolves around solving such an equation using the method of completing the square, which is a fundamental algebraic technique.

💡Coefficient

In algebra, a coefficient is a numerical factor multiplying a variable. In the context of the video, the presenter refers to the coefficient of the 'x' term when demonstrating the process of completing the square, which is a key step in the method.

💡Binomial Squared

A binomial squared refers to the expansion of the square of a binomial expression, such as (x + a)^2. The video script uses this concept to reformat the quadratic equation into a form that allows for taking the square root to solve for 'x'.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The video demonstrates taking the square root of both sides of an equation to simplify it, which is a common technique in solving quadratic equations.

💡Isolate the Variable

Isolating the variable is the process of getting the variable alone on one side of the equation to solve for its value. The video script illustrates this by moving terms around to solve for 'x', which is a fundamental step in algebraic problem-solving.

💡Plus or Minus

In the context of solving equations, 'plus or minus' indicates that there are two possible solutions, one positive and one negative. The video corrects Photomath's error by showing that the correct solutions involve both a positive and negative value, which is a common outcome when dealing with square roots of positive numbers.

Highlights

Photomath gives an incorrect initial answer for a math problem involving completing the square.

The correct answer is not immediately visible in Photomath's solution, prompting a deeper explanation.

The problem involves an equation that Photomath suggests has no real number solutions.

The presenter aims to demonstrate why Photomath's initial answer is incorrect.

The method of completing the square is introduced as the correct approach to solve the equation.

The equation is simplified by dividing all terms by 3.

The constant term is moved to the other side of the equation to prepare for completing the square.

The b term's coefficient is divided by 2 and then squared to complete the square.

The equation is transformed into a binomial squared by adding and subtracting the squared term.

The presenter explains the process of taking the square root of both sides to solve for x.

The square root of a negative number is addressed, indicating the presence of imaginary numbers.

The final answer includes both real and imaginary components, contrary to Photomath's claim of no real solutions.

The presenter corrects the oversight of not including the plus or minus sign in the solution.

The correct final answer is x equals negative 1 plus or minus the square root of 2 times i.

Photomath is shown to be incorrect in its initial assessment of the problem.

The importance of understanding and correctly applying mathematical methods is emphasized.

The presenter encourages a deeper understanding of math problems beyond relying on app solutions.

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